Complex-analytic and other properties of the generalized - PowerPoint PPT Presentation

Complex-analytic and other properties of the generalized hypergeometric functions and their ratios Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Far Eastern Federal University New Developments in Complex Analysis and Function Theory Heraklion, Greece, July 2–6, 2018 Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function

Generalized hypergeometric function: questions Set a = ( a 1 , a 2 , . . . , a p ) ∈ C p , b = ( b 1 , b 2 , . . . , b q ) ∈ C q . Then � a � ∞ � ( a 1 ) n ( a 2 ) n · · · ( a p ) n � � ( b 1 ) n ( b 2 ) n · · · ( b q ) n n ! z n , p F q � z = p F q ( a ; b ; z ) := � b n =0 where ( a ) n = Γ( a + n ) / Γ( a ) denotes the rising factorial. The series converges for all z ∈ C if p ≤ q and for | z | < 1 if p = q + 1 . Analytic continuation of z → p F p − 1 ( z ) to | z | ≥ 1 Analytic continuation of ( a , b ) → p F p − 1 (1) to �� � k a k − � ℜ j b j > 0 ( p F p − 1 (1) is important in physics) Geometric properties of z → p F p − 1 ( z ) (univalence, starlikeness, convexity etc.) and ratios Values of z → p F p − 1 ( z ) on the banks of the branch cut [1 , ∞ ) Bounds for z → p F p − 1 ( z ) in the complex plane Location of zeros of entire functions p F q , p ≤ q (reality of zeros, zero-free regions, etc.) Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function

Important ingredient: Meijer’s G -function Definition of Meijer’s G -function (Meijer, around 1940) Suppose 0 ≤ m ≤ q , 0 ≤ n ≤ p are integers, a = ( a 1 , a 2 , . . . , a p ) ∈ C p , b = ( b 1 , b 2 , . . . , b q ) ∈ C q are such that a i − b j / ∈ N for i = 1 , . . . , n , j = 1 , . . . , m . Define � � a G m,n z := p,q b � Γ( b 1 + s ) · · · Γ( b m + s )Γ(1 − a 1 − s ) · · · Γ(1 − a n − s ) 1 z − s ds. 2 πi Γ( a n +1 + s ) · · · Γ( a p + s )Γ(1 − b m +1 − s ) · · · Γ(1 − b q − s ) L � �� � G ( s ) The contour L begins and ends at infinity and separates the poles − b j − k , k = 0 , 1 , . . . from the poles 1 − a i + l , l = 0 , 1 , . . . Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function

Important ingredient: Meijer’s G -function Mostly, we need only a particular case (Meijer-Nørlund function): � � � 1 Γ( a 1 + s ) · · · Γ( a p + s ) b G p, 0 Γ( b 1 + s ) · · · Γ( b q + s ) z − s ds. z := q,p a 2 πi L The contour L is a vertical line on the right of all the poles of the integrand or the left loop beginning and ending at −∞ and leaving all the poles on the left. Notation: Γ( a ) = Γ( a 1 )Γ( a 2 ) · · · Γ( a p ) , ( a ) n = ( a 1 ) n ( a 2 ) n · · · ( a p ) n , a + µ = ( a 1 + µ, a 2 + µ, . . . , a p + µ ); in particular, ( a ) = ( a ) 1 = a 1 · · · a p ; inequalities like ℜ ( a ) > 0 and properties like − a / ∈ N 0 will be understood element-wise. The symbol a [ k ] stands for the vector a with omitted k -th element. Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function

Key tool: integral representations Termwise integration leads to the Laplace transform representations � dt � a � ∞ � � � � = Γ( b ) b � � e − zt G p +1 , 0 � − z p +1 F p t t , � � p,p +1 b Γ( a ) a � 0 � a � dt � 1 � � � � = Γ( b ) b � � e − zt G p, 0 p F p � − z t t , � � p,p b Γ( a ) a � 0 the generalized Stieltjes transform representation � σ, a � 1 � � � � � = Γ( b ) dt b � � G p, 0 p +1 F p � − z t � � p,p b Γ( a ) a t (1 + zt ) σ � 0 and the cosine Fourier transform representation � 1 � � � a � � � dt 2Γ( b ) b � � � − z 2 / 4 cos( zt ) G p, 0 t 2 p − 1 F p = √ π Γ( a ) t . � � p,p a , 1 / 2 b � 0 These hold if ℜ ( a ) > 0 and also � p i =1 ℜ ( b i − a i ) > 0 in the second and third formulas or � p i =1 ℜ ( b i ) − � p − 1 i =1 ℜ ( a i ) > 1 / 2 in the last formula. Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function

Extended integral representations with atom First appearance: 1994 book by Virginia Kiryakova (derived by consecutive fractional integrations). We relaxed the restrictions on parameters and further extended these formulas to zero parametric excess � p i =1 ( b i − a i ) = 0 as follows � dt � a � 1 � � � � � � = Γ( b ) b � � G p, 0 e − zt p F p � − z t + δ 1 t , � � p,p b Γ( a ) a � 0 � σ, a � 1 � � � � � � � = Γ( b ) dt b � � G p, 0 � − z p +1 F p t + δ 1 t (1 + zt ) σ , � � p,p b Γ( a ) a � 0 � � a � 2Γ( b ) � � − z 2 / 4 p − 1 F p = √ π Γ( a ) × � b � 1 � � � � � dt b � G p, 0 t 2 cos( zt ) + δ 1 t , � p,p a , 1 / 2 � 0 where δ 1 denotes the unit mass at the point t = 1 and � p i =1 b i − � p − 1 i =1 a i = 1 / 2 in the last formula. Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function

Positivity of G -function Proposition (K.-Prilepkina, 2012) Suppose a , b ∈ R p and v a , b ( t ) = � p j =1 ( t a j − t b j ) ≥ 0 . Then � � b G p, 0 t ≥ 0 p,p a on (0 , 1) and � � b G p +1 , 0 t ≥ 0 p,p +1 σ, a on (0 , ∞ ) for any σ > 0 . In fact, more is true: if also a , b > 0 then � � Γ( b ) is infinitely divisible probability e − t b Γ( a ) G p, 0 dt p,p distribution on [0 , ∞ ) . a Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function

Sufficient condition: weak majorization Observation (Alzer (1997) based on Tomi´ c (1949)) : v a , b ( t ) ≥ 0 on [0 , 1] if 0 ≤ a 1 ≤ a 2 ≤ · · · ≤ a p , 0 ≤ b 1 ≤ b 2 ≤ · · · ≤ b p , k k � � and a i ≤ b i for k = 1 , 2 . . . , p. i =1 i =1 These inequalities are known as weak supermajorization and are abbreviated as b ≺ W a , where a =( a 1 , . . . , a p ) , b =( b 1 , . . . , b p ) . Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function

Markov representation Definition: Markov functions Define T to be the class of functions f representable by � 1 dµ ( t ) f ( z ) = 1 − zt 0 for some probability measure µ on [0 , 1] . Functions f ∈ T are generating functions of the Hausdorff moment sequences. Theorem (K.-Prilepkina, 2012) Suppose 0 < σ ≤ 1 , a > 0 and v a , b ( t ) ≥ 0 on [0 , 1] (in particular, it suffices that b ≺ W a ). Then p +1 F p ( σ, a ; b ; z ) ∈ T and the representing measure is given by � dt � Γ( b ) 1 , b Γ( σ )Γ( a ) G p +1 , 0 dµ ( t ) = t p +1 ,p +1 σ, a t if � ( b k − a k ) > 0 or dµ 1 ( t ) = dµ ( t ) + δ 1 if � ( b k − a k ) = 0 . Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function

Corollary 1 (K.-Prilepkina, 2012) Suppose 0 < σ ≤ 1 and v a , b ( t ) ≥ 0 on [0 , 1] . Then the functions z → p +1 F p ( σ, a ; b ; z ) and z → z p +1 F p ( σ, a ; b ; z ) are univalent in the half-plane ℜ ( z ) < 1 . The second function is √ � starlike is the disk | z | < r ∗ , where r ∗ = 13 13 − 46 ≈ 0 , 934 . Corollary 2 (K.-Prilepkina, 2012) Suppose 0 < σ ≤ 2 and v a , b ( t ) ≥ 0 on [0 , 1] . Then the function z → z p +1 F p ( σ, a ; b ; z ) is univalent in the disk � √ | z | < r s := 32 − 5 ≈ 0 . 81 . Corollary 3 (K.-Prilepkina, 2012) Suppose σ ≥ 1 and v a , b ( t ) ≥ 0 on [0 , 1] . Then the function p +1 F p ( σ, a ; b ; − z ) maps the sector 0 < arg( z ) < π/σ into the lower half-plane ℑ ( z ) < 0 . Dmitrii Karp (joint work with Elena Prilepkina, Jos´ e Luis L´ opez, Alexander Dyachenko and Sergei Kalmykov) dmkrp.wordpress.com Generalized hypergeometric function