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Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Outline Introduction 1 Orthogonal polynomials An example (key idea) Another example (key idea) Multiplicative Renormalization Method 2 OP-generating function MRM procedure Classical distributions Characterization Theorems 3 Characterization problems MRM-applicable measures MRM-factors References 4 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Let µ be Poisson with parameter λ . Then Ee ρ ( t ) x = e − λ ( 1 − e ρ ( t ) ) ψ ( t , x ) := e ρ ( t ) x Ee ρ ( t ) x = e ρ ( t ) x + λ ( 1 − e ρ ( t ) ) Then we have � � = e λ ( e ρ ( t ) − 1 )( e ρ ( s ) − 1 ) E ψ ( t , x ) ψ ( s , x ) � � Observation E ψ ( t , x ) ψ ( s , x ) is a function of ts if we take e ρ ( t ) − 1 = t , i . e ., ρ ( t ) = ln ( 1 + t ) Thus we have the function ψ ( t , x ) = e − λ t ( 1 + t ) x Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Let µ be Poisson with parameter λ . Then Ee ρ ( t ) x = e − λ ( 1 − e ρ ( t ) ) ψ ( t , x ) := e ρ ( t ) x Ee ρ ( t ) x = e ρ ( t ) x + λ ( 1 − e ρ ( t ) ) Then we have � � = e λ ( e ρ ( t ) − 1 )( e ρ ( s ) − 1 ) E ψ ( t , x ) ψ ( s , x ) � � Observation E ψ ( t , x ) ψ ( s , x ) is a function of ts if we take e ρ ( t ) − 1 = t , i . e ., ρ ( t ) = ln ( 1 + t ) Thus we have the function ψ ( t , x ) = e − λ t ( 1 + t ) x Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Let µ be Poisson with parameter λ . Then Ee ρ ( t ) x = e − λ ( 1 − e ρ ( t ) ) ψ ( t , x ) := e ρ ( t ) x Ee ρ ( t ) x = e ρ ( t ) x + λ ( 1 − e ρ ( t ) ) Then we have � � = e λ ( e ρ ( t ) − 1 )( e ρ ( s ) − 1 ) E ψ ( t , x ) ψ ( s , x ) � � Observation E ψ ( t , x ) ψ ( s , x ) is a function of ts if we take e ρ ( t ) − 1 = t , i . e ., ρ ( t ) = ln ( 1 + t ) Thus we have the function ψ ( t , x ) = e − λ t ( 1 + t ) x Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Let µ be Poisson with parameter λ . Then Ee ρ ( t ) x = e − λ ( 1 − e ρ ( t ) ) ψ ( t , x ) := e ρ ( t ) x Ee ρ ( t ) x = e ρ ( t ) x + λ ( 1 − e ρ ( t ) ) Then we have � � = e λ ( e ρ ( t ) − 1 )( e ρ ( s ) − 1 ) E ψ ( t , x ) ψ ( s , x ) � � Observation E ψ ( t , x ) ψ ( s , x ) is a function of ts if we take e ρ ( t ) − 1 = t , i . e ., ρ ( t ) = ln ( 1 + t ) Thus we have the function ψ ( t , x ) = e − λ t ( 1 + t ) x Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Let µ be Poisson with parameter λ . Then Ee ρ ( t ) x = e − λ ( 1 − e ρ ( t ) ) ψ ( t , x ) := e ρ ( t ) x Ee ρ ( t ) x = e ρ ( t ) x + λ ( 1 − e ρ ( t ) ) Then we have � � = e λ ( e ρ ( t ) − 1 )( e ρ ( s ) − 1 ) E ψ ( t , x ) ψ ( s , x ) � � Observation E ψ ( t , x ) ψ ( s , x ) is a function of ts if we take e ρ ( t ) − 1 = t , i . e ., ρ ( t ) = ln ( 1 + t ) Thus we have the function ψ ( t , x ) = e − λ t ( 1 + t ) x Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Let µ be Poisson with parameter λ . Then Ee ρ ( t ) x = e − λ ( 1 − e ρ ( t ) ) ψ ( t , x ) := e ρ ( t ) x Ee ρ ( t ) x = e ρ ( t ) x + λ ( 1 − e ρ ( t ) ) Then we have � � = e λ ( e ρ ( t ) − 1 )( e ρ ( s ) − 1 ) E ψ ( t , x ) ψ ( s , x ) � � Observation E ψ ( t , x ) ψ ( s , x ) is a function of ts if we take e ρ ( t ) − 1 = t , i . e ., ρ ( t ) = ln ( 1 + t ) Thus we have the function ψ ( t , x ) = e − λ t ( 1 + t ) x Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Expand ψ ( t , x ) as a power series in t : � ∞ 1 n ! C n ( x ) t n ψ ( t , x ) = n = 0 where C n ( x ) is a polynomial given by � n � n � ( − λ ) k p x , n − k C n ( x ) = k k = 0 with p x , 0 = 1 , p x , m = x ( x − 1 )( x − 2 ) · · · ( x − m + 1 ) , m ≥ 1. Key Idea The above Observation = ⇒ C n ’s are orthogonal (Charlier polynomials) Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Expand ψ ( t , x ) as a power series in t : � ∞ 1 n ! C n ( x ) t n ψ ( t , x ) = n = 0 where C n ( x ) is a polynomial given by � n � n � ( − λ ) k p x , n − k C n ( x ) = k k = 0 with p x , 0 = 1 , p x , m = x ( x − 1 )( x − 2 ) · · · ( x − m + 1 ) , m ≥ 1. Key Idea The above Observation = ⇒ C n ’s are orthogonal (Charlier polynomials) Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Orthogonal polynomials Multiplicative Renormalization Method An example (key idea) Characterization Theorems Another example (key idea) References Expand ψ ( t , x ) as a power series in t : � ∞ 1 n ! C n ( x ) t n ψ ( t , x ) = n = 0 where C n ( x ) is a polynomial given by � n � n � ( − λ ) k p x , n − k C n ( x ) = k k = 0 with p x , 0 = 1 , p x , m = x ( x − 1 )( x − 2 ) · · · ( x − m + 1 ) , m ≥ 1. Key Idea The above Observation = ⇒ C n ’s are orthogonal (Charlier polynomials) Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Outline Introduction 1 Orthogonal polynomials An example (key idea) Another example (key idea) Multiplicative Renormalization Method 2 OP-generating function MRM procedure Classical distributions Characterization Theorems 3 Characterization problems MRM-applicable measures MRM-factors References 4 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let µ be a probab measure with infinite support and { P n ( x ) } the orthog polys from the Gram-Schmidt orthog process. Definition A function ψ ( t , x ) is called an OP-generating function for µ if it has the series expansion in t � ∞ c n P n ( x ) t n ψ ( t , x ) = n = 0 where c n � = 0 for all n . Remark ψ ( t , x ) is called a generating function in the literature. It is a close-form function, e.g., 2 σ 2 t 2 ( Gaussian ) , ψ ( t , x ) = e − λ t ( 1 + t ) x ( Poisson ) ψ ( t , x ) = e tx − 1 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let µ be a probab measure with infinite support and { P n ( x ) } the orthog polys from the Gram-Schmidt orthog process. Definition A function ψ ( t , x ) is called an OP-generating function for µ if it has the series expansion in t � ∞ c n P n ( x ) t n ψ ( t , x ) = n = 0 where c n � = 0 for all n . Remark ψ ( t , x ) is called a generating function in the literature. It is a close-form function, e.g., 2 σ 2 t 2 ( Gaussian ) , ψ ( t , x ) = e − λ t ( 1 + t ) x ( Poisson ) ψ ( t , x ) = e tx − 1 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let µ be a probab measure with infinite support and { P n ( x ) } the orthog polys from the Gram-Schmidt orthog process. Definition A function ψ ( t , x ) is called an OP-generating function for µ if it has the series expansion in t � ∞ c n P n ( x ) t n ψ ( t , x ) = n = 0 where c n � = 0 for all n . Remark ψ ( t , x ) is called a generating function in the literature. It is a close-form function, e.g., 2 σ 2 t 2 ( Gaussian ) , ψ ( t , x ) = e − λ t ( 1 + t ) x ( Poisson ) ψ ( t , x ) = e tx − 1 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let µ be a probab measure with infinite support and { P n ( x ) } the orthog polys from the Gram-Schmidt orthog process. Definition A function ψ ( t , x ) is called an OP-generating function for µ if it has the series expansion in t � ∞ c n P n ( x ) t n ψ ( t , x ) = n = 0 where c n � = 0 for all n . Remark ψ ( t , x ) is called a generating function in the literature. It is a close-form function, e.g., 2 σ 2 t 2 ( Gaussian ) , ψ ( t , x ) = e − λ t ( 1 + t ) x ( Poisson ) ψ ( t , x ) = e tx − 1 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References � P n � 2 := λ n = ω 0 ω 1 · · · ω n , n ≥ 0, or ω n = λ n /λ n − 1 Fact Theorem If ψ ( t , x ) is an OP-generating function for µ , then � � ∞ ψ ( t , x ) 2 d µ ( x ) = c 2 n λ n t 2 n R n = 0 � � ∞ � n α n λ n t 2 n + 2 c n c n − 1 λ n t 2 n + 1 � x ψ ( t , x ) 2 d µ ( x ) = c 2 R n = 0 where c − 1 = 0 . Conclusion If we have an OP-generating function ψ ( t , x ) , then we can find { P n ( x ) , α n , ω n } . Question How can we find an OP-generating function ψ ( t , x ) ? Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References � P n � 2 := λ n = ω 0 ω 1 · · · ω n , n ≥ 0, or ω n = λ n /λ n − 1 Fact Theorem If ψ ( t , x ) is an OP-generating function for µ , then � � ∞ ψ ( t , x ) 2 d µ ( x ) = c 2 n λ n t 2 n R n = 0 � � ∞ � n α n λ n t 2 n + 2 c n c n − 1 λ n t 2 n + 1 � x ψ ( t , x ) 2 d µ ( x ) = c 2 R n = 0 where c − 1 = 0 . Conclusion If we have an OP-generating function ψ ( t , x ) , then we can find { P n ( x ) , α n , ω n } . Question How can we find an OP-generating function ψ ( t , x ) ? Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References � P n � 2 := λ n = ω 0 ω 1 · · · ω n , n ≥ 0, or ω n = λ n /λ n − 1 Fact Theorem If ψ ( t , x ) is an OP-generating function for µ , then � � ∞ ψ ( t , x ) 2 d µ ( x ) = c 2 n λ n t 2 n R n = 0 � � ∞ � n α n λ n t 2 n + 2 c n c n − 1 λ n t 2 n + 1 � x ψ ( t , x ) 2 d µ ( x ) = c 2 R n = 0 where c − 1 = 0 . Conclusion If we have an OP-generating function ψ ( t , x ) , then we can find { P n ( x ) , α n , ω n } . Question How can we find an OP-generating function ψ ( t , x ) ? Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References � P n � 2 := λ n = ω 0 ω 1 · · · ω n , n ≥ 0, or ω n = λ n /λ n − 1 Fact Theorem If ψ ( t , x ) is an OP-generating function for µ , then � � ∞ ψ ( t , x ) 2 d µ ( x ) = c 2 n λ n t 2 n R n = 0 � � ∞ � n α n λ n t 2 n + 2 c n c n − 1 λ n t 2 n + 1 � x ψ ( t , x ) 2 d µ ( x ) = c 2 R n = 0 where c − 1 = 0 . Conclusion If we have an OP-generating function ψ ( t , x ) , then we can find { P n ( x ) , α n , ω n } . Question How can we find an OP-generating function ψ ( t , x ) ? Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Outline Introduction 1 Orthogonal polynomials An example (key idea) Another example (key idea) Multiplicative Renormalization Method 2 OP-generating function MRM procedure Classical distributions Characterization Theorems 3 Characterization problems MRM-applicable measures MRM-factors References 4 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let h ( x ) be a “good" function. Define two functions � � � θ ( t ) = h ( tx ) d µ ( x ) , θ ( t , s ) = h ( tx ) h ( sx ) d µ ( x ) R R Theorem (Asai-Kubo-K, TJM 2003) Let ρ ( t ) be an analytic function at 0 with ρ ( 0 ) = 0 and ρ ′ ( 0 ) � = 0 . Then the multiplicative renormalization ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ if and only if the function � θ ( ρ ( t ) , ρ ( s )) Θ ρ ( t , s ) := θ ( ρ ( t )) θ ( ρ ( s )) defined in some neighborhood of ( 0 , 0 ) is a function of ts. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let h ( x ) be a “good" function. Define two functions � � � θ ( t ) = h ( tx ) d µ ( x ) , θ ( t , s ) = h ( tx ) h ( sx ) d µ ( x ) R R Theorem (Asai-Kubo-K, TJM 2003) Let ρ ( t ) be an analytic function at 0 with ρ ( 0 ) = 0 and ρ ′ ( 0 ) � = 0 . Then the multiplicative renormalization ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ if and only if the function � θ ( ρ ( t ) , ρ ( s )) Θ ρ ( t , s ) := θ ( ρ ( t )) θ ( ρ ( s )) defined in some neighborhood of ( 0 , 0 ) is a function of ts. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let h ( x ) be a “good" function. Define two functions � � � θ ( t ) = h ( tx ) d µ ( x ) , θ ( t , s ) = h ( tx ) h ( sx ) d µ ( x ) R R Theorem (Asai-Kubo-K, TJM 2003) Let ρ ( t ) be an analytic function at 0 with ρ ( 0 ) = 0 and ρ ′ ( 0 ) � = 0 . Then the multiplicative renormalization ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ if and only if the function � θ ( ρ ( t ) , ρ ( s )) Θ ρ ( t , s ) := θ ( ρ ( t )) θ ( ρ ( s )) defined in some neighborhood of ( 0 , 0 ) is a function of ts. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let h ( x ) be a “good" function. Define two functions � � � θ ( t ) = h ( tx ) d µ ( x ) , θ ( t , s ) = h ( tx ) h ( sx ) d µ ( x ) R R Theorem (Asai-Kubo-K, TJM 2003) Let ρ ( t ) be an analytic function at 0 with ρ ( 0 ) = 0 and ρ ′ ( 0 ) � = 0 . Then the multiplicative renormalization ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ if and only if the function � θ ( ρ ( t ) , ρ ( s )) Θ ρ ( t , s ) := θ ( ρ ( t )) θ ( ρ ( s )) defined in some neighborhood of ( 0 , 0 ) is a function of ts. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let h ( x ) be a “good" function. Define two functions � � � θ ( t ) = h ( tx ) d µ ( x ) , θ ( t , s ) = h ( tx ) h ( sx ) d µ ( x ) R R Theorem (Asai-Kubo-K, TJM 2003) Let ρ ( t ) be an analytic function at 0 with ρ ( 0 ) = 0 and ρ ′ ( 0 ) � = 0 . Then the multiplicative renormalization ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ if and only if the function � θ ( ρ ( t ) , ρ ( s )) Θ ρ ( t , s ) := θ ( ρ ( t )) θ ( ρ ( s )) defined in some neighborhood of ( 0 , 0 ) is a function of ts. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Let h ( x ) be a “good" function. Define two functions � � � θ ( t ) = h ( tx ) d µ ( x ) , θ ( t , s ) = h ( tx ) h ( sx ) d µ ( x ) R R Theorem (Asai-Kubo-K, TJM 2003) Let ρ ( t ) be an analytic function at 0 with ρ ( 0 ) = 0 and ρ ′ ( 0 ) � = 0 . Then the multiplicative renormalization ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ if and only if the function � θ ( ρ ( t ) , ρ ( s )) Θ ρ ( t , s ) := θ ( ρ ( t )) θ ( ρ ( s )) defined in some neighborhood of ( 0 , 0 ) is a function of ts. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Definition A probab measure µ is called MRM-applicable for h ( x ) if there exists an analytic function ρ ( t ) at 0 with ρ ( 0 ) = 0 , ρ ′ ( 0 ) � = 0 such that ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ . Remark A probab measure can be MRM-applicable for two different functions (not so obvious). Definition A function h ( x ) is called an MRM-factor for µ if µ is MRM-applicable for h ( x ) . Remark A function can be an MRM-factor for several different probab measures (obvious, e.g., e x is an MRM-factor for Gaussian and Poisson measures). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Definition A probab measure µ is called MRM-applicable for h ( x ) if there exists an analytic function ρ ( t ) at 0 with ρ ( 0 ) = 0 , ρ ′ ( 0 ) � = 0 such that ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ . Remark A probab measure can be MRM-applicable for two different functions (not so obvious). Definition A function h ( x ) is called an MRM-factor for µ if µ is MRM-applicable for h ( x ) . Remark A function can be an MRM-factor for several different probab measures (obvious, e.g., e x is an MRM-factor for Gaussian and Poisson measures). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Definition A probab measure µ is called MRM-applicable for h ( x ) if there exists an analytic function ρ ( t ) at 0 with ρ ( 0 ) = 0 , ρ ′ ( 0 ) � = 0 such that ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ . Remark A probab measure can be MRM-applicable for two different functions (not so obvious). Definition A function h ( x ) is called an MRM-factor for µ if µ is MRM-applicable for h ( x ) . Remark A function can be an MRM-factor for several different probab measures (obvious, e.g., e x is an MRM-factor for Gaussian and Poisson measures). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Definition A probab measure µ is called MRM-applicable for h ( x ) if there exists an analytic function ρ ( t ) at 0 with ρ ( 0 ) = 0 , ρ ′ ( 0 ) � = 0 such that ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ . Remark A probab measure can be MRM-applicable for two different functions (not so obvious). Definition A function h ( x ) is called an MRM-factor for µ if µ is MRM-applicable for h ( x ) . Remark A function can be an MRM-factor for several different probab measures (obvious, e.g., e x is an MRM-factor for Gaussian and Poisson measures). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Definition A probab measure µ is called MRM-applicable for h ( x ) if there exists an analytic function ρ ( t ) at 0 with ρ ( 0 ) = 0 , ρ ′ ( 0 ) � = 0 such that ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ . Remark A probab measure can be MRM-applicable for two different functions (not so obvious). Definition A function h ( x ) is called an MRM-factor for µ if µ is MRM-applicable for h ( x ) . Remark A function can be an MRM-factor for several different probab measures (obvious, e.g., e x is an MRM-factor for Gaussian and Poisson measures). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Definition A probab measure µ is called MRM-applicable for h ( x ) if there exists an analytic function ρ ( t ) at 0 with ρ ( 0 ) = 0 , ρ ′ ( 0 ) � = 0 such that ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ . Remark A probab measure can be MRM-applicable for two different functions (not so obvious). Definition A function h ( x ) is called an MRM-factor for µ if µ is MRM-applicable for h ( x ) . Remark A function can be an MRM-factor for several different probab measures (obvious, e.g., e x is an MRM-factor for Gaussian and Poisson measures). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Definition A probab measure µ is called MRM-applicable for h ( x ) if there exists an analytic function ρ ( t ) at 0 with ρ ( 0 ) = 0 , ρ ′ ( 0 ) � = 0 such that ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ . Remark A probab measure can be MRM-applicable for two different functions (not so obvious). Definition A function h ( x ) is called an MRM-factor for µ if µ is MRM-applicable for h ( x ) . Remark A function can be an MRM-factor for several different probab measures (obvious, e.g., e x is an MRM-factor for Gaussian and Poisson measures). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Definition A probab measure µ is called MRM-applicable for h ( x ) if there exists an analytic function ρ ( t ) at 0 with ρ ( 0 ) = 0 , ρ ′ ( 0 ) � = 0 such that ψ ( t , x ) := h ( ρ ( t ) x ) θ ( ρ ( t )) is an OP-generating function for µ . Remark A probab measure can be MRM-applicable for two different functions (not so obvious). Definition A function h ( x ) is called an MRM-factor for µ if µ is MRM-applicable for h ( x ) . Remark A function can be an MRM-factor for several different probab measures (obvious, e.g., e x is an MRM-factor for Gaussian and Poisson measures). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Summary MRM procedure: � � � � θ ( t ) , � µ − → h ( x ) − → θ ( t , s ) − → ρ ( t ) , Θ ρ ( t , s ) − → ψ ( t , x ) Remarks 1. h ( x ) : e x , ( 1 − x ) − κ , hypergeometric functions 2. θ ( t ) =? ( µ given or unknown) 3. ρ ( t ) =? ( µ given or unknown) Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Summary MRM procedure: � � � � θ ( t ) , � µ − → h ( x ) − → θ ( t , s ) − → ρ ( t ) , Θ ρ ( t , s ) − → ψ ( t , x ) Remarks 1. h ( x ) : e x , ( 1 − x ) − κ , hypergeometric functions 2. θ ( t ) =? ( µ given or unknown) 3. ρ ( t ) =? ( µ given or unknown) Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Summary MRM procedure: � � � � θ ( t ) , � µ − → h ( x ) − → θ ( t , s ) − → ρ ( t ) , Θ ρ ( t , s ) − → ψ ( t , x ) Remarks 1. h ( x ) : e x , ( 1 − x ) − κ , hypergeometric functions 2. θ ( t ) =? ( µ given or unknown) 3. ρ ( t ) =? ( µ given or unknown) Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Summary MRM procedure: � � � � θ ( t ) , � µ − → h ( x ) − → θ ( t , s ) − → ρ ( t ) , Θ ρ ( t , s ) − → ψ ( t , x ) Remarks 1. h ( x ) : e x , ( 1 − x ) − κ , hypergeometric functions 2. θ ( t ) =? ( µ given or unknown) 3. ρ ( t ) =? ( µ given or unknown) Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Summary MRM procedure: � � � � θ ( t ) , � µ − → h ( x ) − → θ ( t , s ) − → ρ ( t ) , Θ ρ ( t , s ) − → ψ ( t , x ) Remarks 1. h ( x ) : e x , ( 1 − x ) − κ , hypergeometric functions 2. θ ( t ) =? ( µ given or unknown) 3. ρ ( t ) =? ( µ given or unknown) Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Outline Introduction 1 Orthogonal polynomials An example (key idea) Another example (key idea) Multiplicative Renormalization Method 2 OP-generating function MRM procedure Classical distributions Characterization Theorems 3 Characterization problems MRM-applicable measures MRM-factors References 4 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction OP-generating function Multiplicative Renormalization Method MRM procedure Characterization Theorems Classical distributions References Classical distributions with their OP-generating functions: µ h ( x ) θ ( t ) ρ ( t ) ψ ( t , x ) 1 2 σ 2 t 2 e tx − 1 2 σ 2 t 2 e x e t Gaussian e λ ( e t − 1 ) e x e − λ t ( 1 + t ) x Poisson ln ( 1 + t ) tx 1 t e x ( 1 + t ) − α e gamma 1 + t ( 1 − t ) α 1 + t 1 2 2 t √ 1 uniform √ √ √ 1 + t 2 1 − x 1 + t + 1 − t 1 − 2 tx + t 2 1 − t 2 1 √ 1 2 t arcsine 1 − x 1 + t 2 1 − 2 tx + t 2 1 − t 2 1 2 2 t 1 1 + √ semi-circle 1 − x 1 + t 2 1 − 2 tx + t 2 1 − t 2 2 β 1 2 t 1 ( 1 + √ beta ( 1 − x ) β 1 + t 2 ( 1 − 2 tx + t 2 ) β 1 − t 2 ) β ( 1 − q ) r ln 1 + t e x ( 1 + t ) x ( 1 + qt ) − x − r Pascal ( 1 − qe t ) r 1 + qt e x tan − 1 t tan − 1 t e x √ sec t Stoch area 1 + t 2 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Outline Introduction 1 Orthogonal polynomials An example (key idea) Another example (key idea) Multiplicative Renormalization Method 2 OP-generating function MRM procedure Classical distributions Characterization Theorems 3 Characterization problems MRM-applicable measures MRM-factors References 4 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Note that in the previous chart, there are several distributions µ which have the same MRM-factor h ( x ) . Example 1 Gaussian, Poisson, gamma, Pascal, and stochastic area all have the same MRM-factor h ( x ) = e x . Example 2 The arcsine, semi-circle and the beta with β = 1 all have the same MRM-factor h ( x ) = ( 1 − x ) − 1 . This leads to the following characterization problem. First Characterization Problem. Given an MRM-factor h ( x ) , find all probab measures µ which are MRM-applicable for h ( x ) . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Note that in the previous chart, there are several distributions µ which have the same MRM-factor h ( x ) . Example 1 Gaussian, Poisson, gamma, Pascal, and stochastic area all have the same MRM-factor h ( x ) = e x . Example 2 The arcsine, semi-circle and the beta with β = 1 all have the same MRM-factor h ( x ) = ( 1 − x ) − 1 . This leads to the following characterization problem. First Characterization Problem. Given an MRM-factor h ( x ) , find all probab measures µ which are MRM-applicable for h ( x ) . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Note that in the previous chart, there are several distributions µ which have the same MRM-factor h ( x ) . Example 1 Gaussian, Poisson, gamma, Pascal, and stochastic area all have the same MRM-factor h ( x ) = e x . Example 2 The arcsine, semi-circle and the beta with β = 1 all have the same MRM-factor h ( x ) = ( 1 − x ) − 1 . This leads to the following characterization problem. First Characterization Problem. Given an MRM-factor h ( x ) , find all probab measures µ which are MRM-applicable for h ( x ) . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Note that in the previous chart, there are several distributions µ which have the same MRM-factor h ( x ) . Example 1 Gaussian, Poisson, gamma, Pascal, and stochastic area all have the same MRM-factor h ( x ) = e x . Example 2 The arcsine, semi-circle and the beta with β = 1 all have the same MRM-factor h ( x ) = ( 1 − x ) − 1 . This leads to the following characterization problem. First Characterization Problem. Given an MRM-factor h ( x ) , find all probab measures µ which are MRM-applicable for h ( x ) . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Note that in the previous chart, there are several distributions µ which have the same MRM-factor h ( x ) . Example 1 Gaussian, Poisson, gamma, Pascal, and stochastic area all have the same MRM-factor h ( x ) = e x . Example 2 The arcsine, semi-circle and the beta with β = 1 all have the same MRM-factor h ( x ) = ( 1 − x ) − 1 . This leads to the following characterization problem. First Characterization Problem. Given an MRM-factor h ( x ) , find all probab measures µ which are MRM-applicable for h ( x ) . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Note that in the previous chart, there are several distributions µ which have the same MRM-factor h ( x ) . Example 1 Gaussian, Poisson, gamma, Pascal, and stochastic area all have the same MRM-factor h ( x ) = e x . Example 2 The arcsine, semi-circle and the beta with β = 1 all have the same MRM-factor h ( x ) = ( 1 − x ) − 1 . This leads to the following characterization problem. First Characterization Problem. Given an MRM-factor h ( x ) , find all probab measures µ which are MRM-applicable for h ( x ) . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References On the other hand, a probab measure µ may have two different MRM-factors. Example For the semi-circle measure µ , we have two different MRM-factors, which lead to different OP-generating functions: 1 1. h ( x ) = 1 − x is an MRM-factor for µ . In this case, 2 2 t 1 θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − 2 tx + t 2 1 + 1 2. h ( x ) = ( 1 − x ) 2 is an MRM-factor for µ . In this case, 1 − t 2 2 2 t θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − t 2 + ( 1 − 2 tx + t 2 ) 2 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References On the other hand, a probab measure µ may have two different MRM-factors. Example For the semi-circle measure µ , we have two different MRM-factors, which lead to different OP-generating functions: 1 1. h ( x ) = 1 − x is an MRM-factor for µ . In this case, 2 2 t 1 θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − 2 tx + t 2 1 + 1 2. h ( x ) = ( 1 − x ) 2 is an MRM-factor for µ . In this case, 1 − t 2 2 2 t θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − t 2 + ( 1 − 2 tx + t 2 ) 2 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References On the other hand, a probab measure µ may have two different MRM-factors. Example For the semi-circle measure µ , we have two different MRM-factors, which lead to different OP-generating functions: 1 1. h ( x ) = 1 − x is an MRM-factor for µ . In this case, 2 2 t 1 θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − 2 tx + t 2 1 + 1 2. h ( x ) = ( 1 − x ) 2 is an MRM-factor for µ . In this case, 1 − t 2 2 2 t θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − t 2 + ( 1 − 2 tx + t 2 ) 2 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References On the other hand, a probab measure µ may have two different MRM-factors. Example For the semi-circle measure µ , we have two different MRM-factors, which lead to different OP-generating functions: 1 1. h ( x ) = 1 − x is an MRM-factor for µ . In this case, 2 2 t 1 θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − 2 tx + t 2 1 + 1 2. h ( x ) = ( 1 − x ) 2 is an MRM-factor for µ . In this case, 1 − t 2 2 2 t θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − t 2 + ( 1 − 2 tx + t 2 ) 2 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References On the other hand, a probab measure µ may have two different MRM-factors. Example For the semi-circle measure µ , we have two different MRM-factors, which lead to different OP-generating functions: 1 1. h ( x ) = 1 − x is an MRM-factor for µ . In this case, 2 2 t 1 θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − 2 tx + t 2 1 + 1 2. h ( x ) = ( 1 − x ) 2 is an MRM-factor for µ . In this case, 1 − t 2 2 2 t θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − t 2 + ( 1 − 2 tx + t 2 ) 2 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References On the other hand, a probab measure µ may have two different MRM-factors. Example For the semi-circle measure µ , we have two different MRM-factors, which lead to different OP-generating functions: 1 1. h ( x ) = 1 − x is an MRM-factor for µ . In this case, 2 2 t 1 θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − 2 tx + t 2 1 + 1 2. h ( x ) = ( 1 − x ) 2 is an MRM-factor for µ . In this case, 1 − t 2 2 2 t θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − t 2 + ( 1 − 2 tx + t 2 ) 2 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References On the other hand, a probab measure µ may have two different MRM-factors. Example For the semi-circle measure µ , we have two different MRM-factors, which lead to different OP-generating functions: 1 1. h ( x ) = 1 − x is an MRM-factor for µ . In this case, 2 2 t 1 θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − 2 tx + t 2 1 + 1 2. h ( x ) = ( 1 − x ) 2 is an MRM-factor for µ . In this case, 1 − t 2 2 2 t θ ( t ) = √ 1 − t 2 , ρ ( t ) = 1 + t 2 , ψ ( t , x ) = 1 − t 2 + ( 1 − 2 tx + t 2 ) 2 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References This leads to another characterization problem. Second Characterization Problem. Given an MRM-applicable probab measure µ , find all MRM-factors h ( x ) for µ . 2 t Finally, observe that the function ρ ( t ) = 1 + t 2 is the ρ -function for several probab meassures. Thus we also have the following characterization problem: Third Characterization Problem. Given a ρ -function ρ ( t ) , find all MRM-applicable probab measures µ and MRM-factors h ( x ) , which have the given ρ ( t ) as a ρ -function. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References This leads to another characterization problem. Second Characterization Problem. Given an MRM-applicable probab measure µ , find all MRM-factors h ( x ) for µ . 2 t Finally, observe that the function ρ ( t ) = 1 + t 2 is the ρ -function for several probab meassures. Thus we also have the following characterization problem: Third Characterization Problem. Given a ρ -function ρ ( t ) , find all MRM-applicable probab measures µ and MRM-factors h ( x ) , which have the given ρ ( t ) as a ρ -function. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References This leads to another characterization problem. Second Characterization Problem. Given an MRM-applicable probab measure µ , find all MRM-factors h ( x ) for µ . 2 t Finally, observe that the function ρ ( t ) = 1 + t 2 is the ρ -function for several probab meassures. Thus we also have the following characterization problem: Third Characterization Problem. Given a ρ -function ρ ( t ) , find all MRM-applicable probab measures µ and MRM-factors h ( x ) , which have the given ρ ( t ) as a ρ -function. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References This leads to another characterization problem. Second Characterization Problem. Given an MRM-applicable probab measure µ , find all MRM-factors h ( x ) for µ . 2 t Finally, observe that the function ρ ( t ) = 1 + t 2 is the ρ -function for several probab meassures. Thus we also have the following characterization problem: Third Characterization Problem. Given a ρ -function ρ ( t ) , find all MRM-applicable probab measures µ and MRM-factors h ( x ) , which have the given ρ ( t ) as a ρ -function. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References This leads to another characterization problem. Second Characterization Problem. Given an MRM-applicable probab measure µ , find all MRM-factors h ( x ) for µ . 2 t Finally, observe that the function ρ ( t ) = 1 + t 2 is the ρ -function for several probab meassures. Thus we also have the following characterization problem: Third Characterization Problem. Given a ρ -function ρ ( t ) , find all MRM-applicable probab measures µ and MRM-factors h ( x ) , which have the given ρ ( t ) as a ρ -function. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References This leads to another characterization problem. Second Characterization Problem. Given an MRM-applicable probab measure µ , find all MRM-factors h ( x ) for µ . 2 t Finally, observe that the function ρ ( t ) = 1 + t 2 is the ρ -function for several probab meassures. Thus we also have the following characterization problem: Third Characterization Problem. Given a ρ -function ρ ( t ) , find all MRM-applicable probab measures µ and MRM-factors h ( x ) , which have the given ρ ( t ) as a ρ -function. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Outline Introduction 1 Orthogonal polynomials An example (key idea) Another example (key idea) Multiplicative Renormalization Method 2 OP-generating function MRM procedure Classical distributions Characterization Theorems 3 Characterization problems MRM-applicable measures MRM-factors References 4 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = e x Theorem (Kubo, IDAQP 2004) The class of all MRM-applicale probability measures for the function h ( x ) = e x consists of translations and dilations of Gaussian, Poisson, gamma, Pascal, and Mexiner measures M κ,η with parameter κ > 0 and η ∈ R . Remark The proof of this theorem is relatively easy comparing with other functions h ( x ) . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = e x Theorem (Kubo, IDAQP 2004) The class of all MRM-applicale probability measures for the function h ( x ) = e x consists of translations and dilations of Gaussian, Poisson, gamma, Pascal, and Mexiner measures M κ,η with parameter κ > 0 and η ∈ R . Remark The proof of this theorem is relatively easy comparing with other functions h ( x ) . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = e x Theorem (Kubo, IDAQP 2004) The class of all MRM-applicale probability measures for the function h ( x ) = e x consists of translations and dilations of Gaussian, Poisson, gamma, Pascal, and Mexiner measures M κ,η with parameter κ > 0 and η ∈ R . Remark The proof of this theorem is relatively easy comparing with other functions h ( x ) . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 Note that in the chart of classical distributions, there are two probab measures that are MRM-applicable for h ( x ) = ( 1 − x ) − 1 , namely, (1) Arcsine d µ ( x ) = 1 1 √ 1 − x 2 dx , | x | < 1 , π 1 − t 2 ψ ( t , x ) = 1 − 2 tx + t 2 . (2) Semi-circle � d µ ( x ) = 2 1 − x 2 dx , | x | < 1 , π 1 ψ ( t , x ) = 1 − 2 tx + t 2 . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 Note that in the chart of classical distributions, there are two probab measures that are MRM-applicable for h ( x ) = ( 1 − x ) − 1 , namely, (1) Arcsine d µ ( x ) = 1 1 √ 1 − x 2 dx , | x | < 1 , π 1 − t 2 ψ ( t , x ) = 1 − 2 tx + t 2 . (2) Semi-circle � d µ ( x ) = 2 1 − x 2 dx , | x | < 1 , π 1 ψ ( t , x ) = 1 − 2 tx + t 2 . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 Note that in the chart of classical distributions, there are two probab measures that are MRM-applicable for h ( x ) = ( 1 − x ) − 1 , namely, (1) Arcsine d µ ( x ) = 1 1 √ 1 − x 2 dx , | x | < 1 , π 1 − t 2 ψ ( t , x ) = 1 − 2 tx + t 2 . (2) Semi-circle � d µ ( x ) = 2 1 − x 2 dx , | x | < 1 , π 1 ψ ( t , x ) = 1 − 2 tx + t 2 . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 Note that in the chart of classical distributions, there are two probab measures that are MRM-applicable for h ( x ) = ( 1 − x ) − 1 , namely, (1) Arcsine d µ ( x ) = 1 1 √ 1 − x 2 dx , | x | < 1 , π 1 − t 2 ψ ( t , x ) = 1 − 2 tx + t 2 . (2) Semi-circle � d µ ( x ) = 2 1 − x 2 dx , | x | < 1 , π 1 ψ ( t , x ) = 1 − 2 tx + t 2 . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Question Are there other probab measures that are MRM-applicable for h ( x ) = ( 1 − x ) − 1 ? Theorem (Kubo-Namli-K, IDAQP 2006) For 0 < a ≤ 1 , the probab measure √ 1 − x 2 a a 2 + ( 1 − 2 a ) x 2 � dx , d µ a ( x ) = � | x | < 1 , π is MRM-applicable for h ( x ) = ( 1 − x ) − 1 with OP-generating function given by ψ a ( t , x ) = 1 + ( 1 − 2 a ) t 2 1 − 2 tx + t 2 . Remark (1) semi-circle: a = 1 2 . (2) arcsine: a = 1. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Question Are there other probab measures that are MRM-applicable for h ( x ) = ( 1 − x ) − 1 ? Theorem (Kubo-Namli-K, IDAQP 2006) For 0 < a ≤ 1 , the probab measure √ 1 − x 2 a a 2 + ( 1 − 2 a ) x 2 � dx , d µ a ( x ) = � | x | < 1 , π is MRM-applicable for h ( x ) = ( 1 − x ) − 1 with OP-generating function given by ψ a ( t , x ) = 1 + ( 1 − 2 a ) t 2 1 − 2 tx + t 2 . Remark (1) semi-circle: a = 1 2 . (2) arcsine: a = 1. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Question Are there other probab measures that are MRM-applicable for h ( x ) = ( 1 − x ) − 1 ? Theorem (Kubo-Namli-K, IDAQP 2006) For 0 < a ≤ 1 , the probab measure √ 1 − x 2 a a 2 + ( 1 − 2 a ) x 2 � dx , d µ a ( x ) = � | x | < 1 , π is MRM-applicable for h ( x ) = ( 1 − x ) − 1 with OP-generating function given by ψ a ( t , x ) = 1 + ( 1 − 2 a ) t 2 1 − 2 tx + t 2 . Remark (1) semi-circle: a = 1 2 . (2) arcsine: a = 1. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References The above theorem shows that there is a family { µ a ; 0 < a ≤ 1 } of MRM-applicable probab measures for h ( x ) = ( 1 − x ) − 1 . But are they all? It turns out that there are many more. But the computation is much much harder and more involved. Lemma Let µ be MRM-applicable for h ( x ) = ( 1 − x ) − 1 . Then ρ ( t ) , θ ( ρ ( t )) , and ψ ( t , x ) must be given by 2 t 1 ρ ( t ) = α + 2 β t + γ 2 , θ ( ρ ( t )) = 1 − ( b + at ) ρ ( t ) , ψ ( t , x ) = α + 2 ( β − b ) t + ( γ − 2 a ) t 2 , a − 2 t ( x − β ) + γ t 2 where α, β, γ, a , b are constants under some constraints. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References The above theorem shows that there is a family { µ a ; 0 < a ≤ 1 } of MRM-applicable probab measures for h ( x ) = ( 1 − x ) − 1 . But are they all? It turns out that there are many more. But the computation is much much harder and more involved. Lemma Let µ be MRM-applicable for h ( x ) = ( 1 − x ) − 1 . Then ρ ( t ) , θ ( ρ ( t )) , and ψ ( t , x ) must be given by 2 t 1 ρ ( t ) = α + 2 β t + γ 2 , θ ( ρ ( t )) = 1 − ( b + at ) ρ ( t ) , ψ ( t , x ) = α + 2 ( β − b ) t + ( γ − 2 a ) t 2 , a − 2 t ( x − β ) + γ t 2 where α, β, γ, a , b are constants under some constraints. Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Theorem (Kubo-Namli-K, COSA 2007) For any a > 0 and | b | ≤ 1 − a, the probab meassure √ 1 − x 2 a a 2 + b 2 − 2 b ( 1 − a ) x + ( 1 − 2 a ) x 2 � dx , | x | < 1 , d µ a , b ( x ) = � π is MRM-applicable for h ( x ) = ( 1 − x ) − 1 with OP-generating function given by ψ a , b ( t , x ) = 1 − 2 bt + ( 1 − 2 a ) t 2 . 1 − 2 tx + t 2 Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Theorem (Kubo-Namli-K, COSA 2007) The class of all MRM-applicale probability measures for the function h ( x ) = ( 1 − x ) − 1 consists of translations and dilations of the probab measures of the form √ 1 − x 2 d µ ( x ) = W 0 π ( 1 − px )( 1 − qx ) 1 ( − 1 , 1 ) ( x ) dx + W 1 d δ 1 p ( x ) + W 2 d δ 1 q ( x ) , where δ c is the Dirac delta meassure at c and p , q , W 0 , W 1 , W 2 are constants depending on two parameters A > 0 and B ≥ 0 . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 / 2 In the chart of classical distributions, the uniform distribution is MRM-applicable for the function h ( x ) = ( 1 − x ) − 1 / 2 . Are there other MRM-applicable probab measures for this function? The computation for trying to find out the answer is rather complicated. Ideas ϕ ( t ) := θ ( ρ ( t )) satisfies the Fundamental Equations: ϕ ′ ( t ) ϕ ( t ) = F 1 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 2 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 3 ( ρ ( t ) , ρ ′ ( t ) , t ) which can be solved (extremely complicated!) to find possible forms of ρ ( t ) . Then derive ϕ ( t ) and θ ( t ) , and finally µ . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 / 2 In the chart of classical distributions, the uniform distribution is MRM-applicable for the function h ( x ) = ( 1 − x ) − 1 / 2 . Are there other MRM-applicable probab measures for this function? The computation for trying to find out the answer is rather complicated. Ideas ϕ ( t ) := θ ( ρ ( t )) satisfies the Fundamental Equations: ϕ ′ ( t ) ϕ ( t ) = F 1 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 2 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 3 ( ρ ( t ) , ρ ′ ( t ) , t ) which can be solved (extremely complicated!) to find possible forms of ρ ( t ) . Then derive ϕ ( t ) and θ ( t ) , and finally µ . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 / 2 In the chart of classical distributions, the uniform distribution is MRM-applicable for the function h ( x ) = ( 1 − x ) − 1 / 2 . Are there other MRM-applicable probab measures for this function? The computation for trying to find out the answer is rather complicated. Ideas ϕ ( t ) := θ ( ρ ( t )) satisfies the Fundamental Equations: ϕ ′ ( t ) ϕ ( t ) = F 1 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 2 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 3 ( ρ ( t ) , ρ ′ ( t ) , t ) which can be solved (extremely complicated!) to find possible forms of ρ ( t ) . Then derive ϕ ( t ) and θ ( t ) , and finally µ . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 / 2 In the chart of classical distributions, the uniform distribution is MRM-applicable for the function h ( x ) = ( 1 − x ) − 1 / 2 . Are there other MRM-applicable probab measures for this function? The computation for trying to find out the answer is rather complicated. Ideas ϕ ( t ) := θ ( ρ ( t )) satisfies the Fundamental Equations: ϕ ′ ( t ) ϕ ( t ) = F 1 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 2 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 3 ( ρ ( t ) , ρ ′ ( t ) , t ) which can be solved (extremely complicated!) to find possible forms of ρ ( t ) . Then derive ϕ ( t ) and θ ( t ) , and finally µ . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 / 2 In the chart of classical distributions, the uniform distribution is MRM-applicable for the function h ( x ) = ( 1 − x ) − 1 / 2 . Are there other MRM-applicable probab measures for this function? The computation for trying to find out the answer is rather complicated. Ideas ϕ ( t ) := θ ( ρ ( t )) satisfies the Fundamental Equations: ϕ ′ ( t ) ϕ ( t ) = F 1 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 2 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 3 ( ρ ( t ) , ρ ′ ( t ) , t ) which can be solved (extremely complicated!) to find possible forms of ρ ( t ) . Then derive ϕ ( t ) and θ ( t ) , and finally µ . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 / 2 In the chart of classical distributions, the uniform distribution is MRM-applicable for the function h ( x ) = ( 1 − x ) − 1 / 2 . Are there other MRM-applicable probab measures for this function? The computation for trying to find out the answer is rather complicated. Ideas ϕ ( t ) := θ ( ρ ( t )) satisfies the Fundamental Equations: ϕ ′ ( t ) ϕ ( t ) = F 1 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 2 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 3 ( ρ ( t ) , ρ ′ ( t ) , t ) which can be solved (extremely complicated!) to find possible forms of ρ ( t ) . Then derive ϕ ( t ) and θ ( t ) , and finally µ . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 / 2 In the chart of classical distributions, the uniform distribution is MRM-applicable for the function h ( x ) = ( 1 − x ) − 1 / 2 . Are there other MRM-applicable probab measures for this function? The computation for trying to find out the answer is rather complicated. Ideas ϕ ( t ) := θ ( ρ ( t )) satisfies the Fundamental Equations: ϕ ′ ( t ) ϕ ( t ) = F 1 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 2 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 3 ( ρ ( t ) , ρ ′ ( t ) , t ) which can be solved (extremely complicated!) to find possible forms of ρ ( t ) . Then derive ϕ ( t ) and θ ( t ) , and finally µ . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References • First Characterization problem for h ( x ) = ( 1 − x ) − 1 / 2 In the chart of classical distributions, the uniform distribution is MRM-applicable for the function h ( x ) = ( 1 − x ) − 1 / 2 . Are there other MRM-applicable probab measures for this function? The computation for trying to find out the answer is rather complicated. Ideas ϕ ( t ) := θ ( ρ ( t )) satisfies the Fundamental Equations: ϕ ′ ( t ) ϕ ( t ) = F 1 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 2 ( ρ ( t ) , ρ ′ ( t ) , t ) = F 3 ( ρ ( t ) , ρ ′ ( t ) , t ) which can be solved (extremely complicated!) to find possible forms of ρ ( t ) . Then derive ϕ ( t ) and θ ( t ) , and finally µ . Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Theorem (Kubo-Namli-K, COSA 2008) A probab measure µ ( with infinite support ) is MRM-applicale for h ( x ) = ( 1 − x ) − 1 / 2 if and only if it is a uniform probab measure on an interval. Remark For the function h ( x ) = ( 1 − x ) − 1 , the corresponding class has a lot of probab measures. But for the function h ( x ) = ( 1 − x ) − 1 / 2 , uniform distribution on [ − 1 , 1 ] is the only one (up to translation and dilation). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials

Introduction Characterization problems Multiplicative Renormalization Method MRM-applicable measures Characterization Theorems MRM-factors References Theorem (Kubo-Namli-K, COSA 2008) A probab measure µ ( with infinite support ) is MRM-applicale for h ( x ) = ( 1 − x ) − 1 / 2 if and only if it is a uniform probab measure on an interval. Remark For the function h ( x ) = ( 1 − x ) − 1 , the corresponding class has a lot of probab measures. But for the function h ( x ) = ( 1 − x ) − 1 / 2 , uniform distribution on [ − 1 , 1 ] is the only one (up to translation and dilation). Hui-Hsiung Kuo Recent results on MRM for orthogonal polynomials