The Jensen-Plya Program for the Riemann Hypothesis and Related - PowerPoint PPT Presentation

Hyperbolicity of Jensen polynomials Our Results on RH Hermite Polynomials Definition The (modified) Hermite polynomials { H d ( X ) : d ≥ 0 } are the orthogonal polynomials with respect to µ ( X ) := e − X 2 4 . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Hermite Polynomials Definition The (modified) Hermite polynomials { H d ( X ) : d ≥ 0 } are the orthogonal polynomials with respect to µ ( X ) := e − X 2 4 . Example (The first few Hermite polynomials) H 0 ( X ) = 1 H 1 ( X ) = X H 2 ( X ) = X 2 − 2 H 3 ( X ) = X 3 − 6 X Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Hermite Polynomials Lemma The Hermite polynomials satisfy: Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Hermite Polynomials Lemma The Hermite polynomials satisfy: 1 Each H d ( X ) is hyperbolic with d distinct roots. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Hermite Polynomials Lemma The Hermite polynomials satisfy: 1 Each H d ( X ) is hyperbolic with d distinct roots. 2 If S d denotes the “suitably normalized” zeros of H d ( X ) , then S d − → Wigner’s Semicircle Law. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH RH Criterion and Hermite Polynomials Theorem 1 (Griffin, O, Rolen, Zagier) The renormalized Jensen polynomials � J d,n ( X ) satisfy γ J d,n � lim ( X ) = H d ( X ) . γ n → + ∞ Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH RH Criterion and Hermite Polynomials Theorem 1 (Griffin, O, Rolen, Zagier) The renormalized Jensen polynomials � J d,n ( X ) satisfy γ J d,n � lim ( X ) = H d ( X ) . γ n → + ∞ For each d at most finitely many J d,n ( X ) are not hyperbolic. γ Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Degree 3 Normalized Jensen polynomials J 3 ,n � n ( X ) γ ≈ 0 . 9769 X 3 + 0 . 7570 X 2 − 5 . 8690 X − 1 . 2661 100 ≈ 0 . 9872 X 3 + 0 . 5625 X 2 − 5 . 9153 X − 0 . 9159 200 ≈ 0 . 9911 X 3 + 0 . 4705 X 2 − 5 . 9374 X − 0 . 7580 300 ≈ 0 . 9931 X 3 + 0 . 4136 X 2 − 5 . 9501 X − 0 . 6623 400 . . . . . . ≈ 0 . 9999 X 3 + 0 . 0009 X 2 − 5 . 9999 X − 0 . 0014 10 8 . . . . . . H 3 ( X ) = X 3 − 6 X ∞ Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Random Matrix Model Predictions Freeman Dyson Hugh Montgomery Andrew Odlyzko Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Random Matrix Model Predictions Freeman Dyson Hugh Montgomery Andrew Odlyzko Gaussian Unitary Ensemble (GUE) (1970s) The nontrivial zeros of ζ ( s ) appear to be “distributed like” the eigenvalues of random Hermitian matrices. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Relation to our work “Theorem” (Griffin, O, Rolen, Zagier) GUE holds for Riemann’s ζ ( s ) in derivative aspect . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Relation to our work “Theorem” (Griffin, O, Rolen, Zagier) GUE holds for Riemann’s ζ ( s ) in derivative aspect . Sketch of Proof 1 The J d,n ( X ) model the zeros of the n th derivative Ξ ( n ) ( X ) . γ Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Relation to our work “Theorem” (Griffin, O, Rolen, Zagier) GUE holds for Riemann’s ζ ( s ) in derivative aspect . Sketch of Proof 1 The J d,n ( X ) model the zeros of the n th derivative Ξ ( n ) ( X ) . γ 2 The derivatives are predicted to satisfy GUE. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Relation to our work “Theorem” (Griffin, O, Rolen, Zagier) GUE holds for Riemann’s ζ ( s ) in derivative aspect . Sketch of Proof 1 The J d,n ( X ) model the zeros of the n th derivative Ξ ( n ) ( X ) . γ 2 The derivatives are predicted to satisfy GUE. 3 For fixed d , we proved that J d,n � lim ( X ) = H d ( X ) . γ n → + ∞ Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Relation to our work “Theorem” (Griffin, O, Rolen, Zagier) GUE holds for Riemann’s ζ ( s ) in derivative aspect . Sketch of Proof 1 The J d,n ( X ) model the zeros of the n th derivative Ξ ( n ) ( X ) . γ 2 The derivatives are predicted to satisfy GUE. 3 For fixed d , we proved that J d,n � lim ( X ) = H d ( X ) . γ n → + ∞ 4 The zeros of the { H d ( X ) } and the eigenvalues in GUE both satisfy Wigner’s Semicircle Distribution. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Computing derivatives Is not Easy Theorem (Pustylnikov (2001), Coffey (2009)) As n → + ∞ , we have Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Computing derivatives Is not Easy Theorem (Pustylnikov (2001), Coffey (2009)) As n → + ∞ , we have Remarks 1 Derivatives essentially drop to 0 for “small” n before exhibiting exponential growth. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Computing derivatives Is not Easy Theorem (Pustylnikov (2001), Coffey (2009)) As n → + ∞ , we have Remarks 1 Derivatives essentially drop to 0 for “small” n before exhibiting exponential growth. 2 This is insufficient for approximating J d,n ( X ) . γ Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH First 10 Taylor coefficients of Ξ( x ) Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Arbitrary precision asymptotics for Ξ (2 n ) (0) Notation 1 We let θ 0 ( t ) := � ∞ k =1 e − πk 2 t , Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Arbitrary precision asymptotics for Ξ (2 n ) (0) Notation 1 We let θ 0 ( t ) := � ∞ k =1 e − πk 2 t , and define � ∞ (log t ) n t − 3 / 4 θ 0 ( t ) dt. F ( n ) := 1 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Arbitrary precision asymptotics for Ξ (2 n ) (0) Notation 1 We let θ 0 ( t ) := � ∞ k =1 e − πk 2 t , and define � ∞ (log t ) n t − 3 / 4 θ 0 ( t ) dt. F ( n ) := 1 2 Following Riemann, we have � n � Ξ ( n ) (0) = ( − 1) n/ 2 · 32 F ( n − 2) − F ( n ) 2 2 n +2 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Arbitrary precision asymptotics for Ξ (2 n ) (0) Notation 1 We let θ 0 ( t ) := � ∞ k =1 e − πk 2 t , and define � ∞ (log t ) n t − 3 / 4 θ 0 ( t ) dt. F ( n ) := 1 2 Following Riemann, we have � n � Ξ ( n ) (0) = ( − 1) n/ 2 · 32 F ( n − 2) − F ( n ) 2 2 n +2 � � n 3 Let L = L ( n ) ≈ log be the unique positive solution of log n the equation n = L · ( πe L + 3 4 ) . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Arbitrary precision asymptotics Theorem (Griffin, O, Rolen, Zagier) To all orders , as n → + ∞ , there are b k ∈ Q ( L ) such that e L/ 4 − n/L +3 / 4 � � √ L n +1 1+ b 1 n + b 2 F ( n ) ∼ 2 π � n 2 + · · · , (1 + L ) n − 3 4 L 2 where b 1 = 2 L 4 +9 L 3 +16 L 2 +6 L +2 . 24 ( L +1) 3 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Arbitrary precision asymptotics Theorem (Griffin, O, Rolen, Zagier) To all orders , as n → + ∞ , there are b k ∈ Q ( L ) such that e L/ 4 − n/L +3 / 4 � � √ L n +1 1+ b 1 n + b 2 F ( n ) ∼ 2 π � n 2 + · · · , (1 + L ) n − 3 4 L 2 where b 1 = 2 L 4 +9 L 3 +16 L 2 +6 L +2 . 24 ( L +1) 3 Remarks 1 Using two terms (i.e. b 1 ) suffices for our RH application. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Arbitrary precision asymptotics Theorem (Griffin, O, Rolen, Zagier) To all orders , as n → + ∞ , there are b k ∈ Q ( L ) such that e L/ 4 − n/L +3 / 4 � � √ L n +1 1+ b 1 n + b 2 F ( n ) ∼ 2 π � n 2 + · · · , (1 + L ) n − 3 4 L 2 where b 1 = 2 L 4 +9 L 3 +16 L 2 +6 L +2 . 24 ( L +1) 3 Remarks 1 Using two terms (i.e. b 1 ) suffices for our RH application. ⇒ hyperbolicity for d ≤ 10 20 . 2 Analysis + Computer = Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH Example: � γ ( n ) := two-term approximation Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH How do these asymptotics imply Theorem 1? Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Our Results on RH How do these asymptotics imply Theorem 1? Theorem 1 is an example of a general phenomenon ! Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Hyperbolic Polynomials in Mathematics Remark Hyperbolicity of “generating polynomials” is studied in enumerative combinatorics in connection with log-concavity a ( n ) 2 ≥ a ( n − 1) a ( n + 1) . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Hyperbolic Polynomials in Mathematics Remark Hyperbolicity of “generating polynomials” is studied in enumerative combinatorics in connection with log-concavity a ( n ) 2 ≥ a ( n − 1) a ( n + 1) . Group theory (lattice subgroup enumeration) Graph theory Symmetric functions Additive number theory (partitions) . . . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Appropriate Growth Definition A real sequence a ( n ) has appropriate growth if Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Appropriate Growth Definition A real sequence a ( n ) has appropriate growth if a ( n + j ) ∼ a ( n ) e A ( n ) j − δ ( n ) 2 j 2 ( n → + ∞ ) for each j for real sequences { A ( n ) } and { δ ( n ) } → 0 . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Appropriate Growth Definition A real sequence a ( n ) has appropriate growth if a ( n + j ) ∼ a ( n ) e A ( n ) j − δ ( n ) 2 j 2 ( n → + ∞ ) for each j for real sequences { A ( n ) } and { δ ( n ) } → 0 . What do we mean? For fixed d and 0 ≤ j ≤ d , as n → + ∞ we have Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Appropriate Growth Definition A real sequence a ( n ) has appropriate growth if a ( n + j ) ∼ a ( n ) e A ( n ) j − δ ( n ) 2 j 2 ( n → + ∞ ) for each j for real sequences { A ( n ) } and { δ ( n ) } → 0 . What do we mean? For fixed d and 0 ≤ j ≤ d , as n → + ∞ we have � a ( n + j ) � log a ( n ) � δ ( n ) d +1 � d � = A ( n ) j − δ ( n ) 2 j 2 + o i,d ( δ ( n ) i ) j i + O d . i =0 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions General Theorem Definition If a ( n ) has appropriate growth, then the renormalized Jensen polynomials are defined by � δ ( n ) X − 1 � 1 J d,n � a ( n ) · δ ( n ) d · J d,n ( X ) := . a a exp( A ( n )) Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions General Theorem Definition If a ( n ) has appropriate growth, then the renormalized Jensen polynomials are defined by � δ ( n ) X − 1 � 1 J d,n � a ( n ) · δ ( n ) d · J d,n ( X ) := . a a exp( A ( n )) General Theorem (Griffin, O, Rolen, Zagier) Suppose that a ( n ) has appropriate growth . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions General Theorem Definition If a ( n ) has appropriate growth, then the renormalized Jensen polynomials are defined by � δ ( n ) X − 1 � 1 J d,n � a ( n ) · δ ( n ) d · J d,n ( X ) := . a a exp( A ( n )) General Theorem (Griffin, O, Rolen, Zagier) Suppose that a ( n ) has appropriate growth . For each degree d ≥ 1 we have J d,n � lim ( X ) = H d ( X ) . a n → + ∞ Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions General Theorem Definition If a ( n ) has appropriate growth, then the renormalized Jensen polynomials are defined by � δ ( n ) X − 1 � 1 J d,n � a ( n ) · δ ( n ) d · J d,n ( X ) := . a a exp( A ( n )) General Theorem (Griffin, O, Rolen, Zagier) Suppose that a ( n ) has appropriate growth . For each degree d ≥ 1 we have J d,n � lim ( X ) = H d ( X ) . a n → + ∞ For each d at most finitely many J d,n ( X ) are not hyperbolic. a Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Motivation for our work Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Motivation for our work Definition A partition is any nonincreasing sequence of integers. p ( n ) := # partitions of size n. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Motivation for our work Definition A partition is any nonincreasing sequence of integers. p ( n ) := # partitions of size n. Example We have that p (4) = 5 because the partitions of 4 are 4 , 3 + 1 , 2 + 2 , 2 + 1 + 1 , 1 + 1 + 1 + 1 . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Log concavity of p ( n ) Example The roots of the quadratic J 2 ,n ( X ) are p � p ( n + 1) 2 − p ( n ) p ( n + 2) − p ( n + 1) ± . p ( n + 2) It is hyperbolic if and only if p ( n + 1) 2 > p ( n ) p ( n + 2) . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Log concavity of p ( n ) Example The roots of the quadratic J 2 ,n ( X ) are p � p ( n + 1) 2 − p ( n ) p ( n + 2) − p ( n + 1) ± . p ( n + 2) It is hyperbolic if and only if p ( n + 1) 2 > p ( n ) p ( n + 2) . Theorem (Nicolas (1978), DeSalvo and Pak (2013)) If n ≥ 25 , then J 2 ,n ( X ) is hyperbolic. p Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Chen’s Conjecture Theorem (Chen, Jia, Wang (2017)) If n ≥ 94 , then J 3 ,n ( X ) is hyperbolic. p Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Chen’s Conjecture Theorem (Chen, Jia, Wang (2017)) If n ≥ 94 , then J 3 ,n ( X ) is hyperbolic. p Conjecture (Chen) There is an N ( d ) where J d,n ( X ) is hyperbolic for all n ≥ N ( d ) . p Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Chen’s Conjecture Theorem (Chen, Jia, Wang (2017)) If n ≥ 94 , then J 3 ,n ( X ) is hyperbolic. p Conjecture (Chen) There is an N ( d ) where J d,n ( X ) is hyperbolic for all n ≥ N ( d ) . p Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Our result Theorem 2 (Griffin, O, Rolen, Zagier) Chen’s Conjecture is true. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Our result Theorem 2 (Griffin, O, Rolen, Zagier) Chen’s Conjecture is true. Remarks 1 The proof can be refined case-by-case to prove the minimality of the claimed N ( d ) (Larson, Wagner). Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Hermite Distributions Another Application Our result Theorem 2 (Griffin, O, Rolen, Zagier) Chen’s Conjecture is true. Remarks 1 The proof can be refined case-by-case to prove the minimality of the claimed N ( d ) (Larson, Wagner). 2 This is a consequence of the General Theorem . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Applications to modular forms Modular forms Definition A weight k weakly holomorphic modular form is a function f on H satisfying: Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Applications to modular forms Modular forms Definition A weight k weakly holomorphic modular form is a function f on H satisfying: For all ( a b c d ) ∈ SL 2 ( Z ) we have 1 � aτ + b � = ( cτ + d ) k f ( τ ) . f cτ + d Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Applications to modular forms Modular forms Definition A weight k weakly holomorphic modular form is a function f on H satisfying: For all ( a b c d ) ∈ SL 2 ( Z ) we have 1 � aτ + b � = ( cτ + d ) k f ( τ ) . f cτ + d The poles of f (if any) are at the cusp ∞ . 2 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Applications to modular forms Modular forms Definition A weight k weakly holomorphic modular form is a function f on H satisfying: For all ( a b c d ) ∈ SL 2 ( Z ) we have 1 � aτ + b � = ( cτ + d ) k f ( τ ) . f cτ + d The poles of f (if any) are at the cusp ∞ . 2 Example (Partition Generating Function) We have the weight − 1 / 2 modular form ∞ p ( n ) e 2 πiτ ( n − 1 � 24 ) . f ( τ ) = n =0 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Applications to modular forms Jensen polynomials for modular forms Theorem 3 (Griffin, O, Rolen, Zagier) Let f be a weakly holomorphic modular form on SL 2 ( Z ) with real coefficients and a pole at i ∞ . Then for each degree d ≥ 1 J d,n � lim a f ( X ) = H d ( X ) . n → + ∞ For each d at most finitely many J d,n a f ( X ) are not hyperbolic. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Applications to modular forms Jensen polynomials for modular forms Theorem 3 (Griffin, O, Rolen, Zagier) Let f be a weakly holomorphic modular form on SL 2 ( Z ) with real coefficients and a pole at i ∞ . Then for each degree d ≥ 1 J d,n � lim a f ( X ) = H d ( X ) . n → + ∞ For each d at most finitely many J d,n a f ( X ) are not hyperbolic. Sketch of Proof. Sufficient asymptotics are known for a f ( n ) in terms of Kloosterman sums and Bessel functions. Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem Natural Questions Question What is special about the Hermite polynomials? Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem Natural Questions Question What is special about the Hermite polynomials? Question Is there an even more general theorem? Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem Hermite Polynomial Generating Function Lemma (Generating Function) We have that � ∞ H d ( X ) · t d d ! = 1 + X · t + ( X 2 − 2) · t 2 e − t 2 + Xt =: 2 + . . . d =0 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem Hermite Polynomial Generating Function Lemma (Generating Function) We have that � ∞ H d ( X ) · t d d ! = 1 + X · t + ( X 2 − 2) · t 2 e − t 2 + Xt =: 2 + . . . d =0 Remark The rough idea of the proof is to show for large fixed n that � ∞ ( X ) · t d d ! ≈ e − t 2 + Xt = e − t 2 · e Xt . J d,n � a d =0 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem More General Theorem Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem More General Theorem Definition A real sequence a ( n ) has appropriate growth for a formal power series F ( t ) := � ∞ i =0 c i t i if Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem More General Theorem Definition A real sequence a ( n ) has appropriate growth for a formal power series F ( t ) := � ∞ i =0 c i t i if a ( n + j ) ∼ a ( n ) E ( n ) j F ( δ ( n ) j ) ( n → + ∞ ) Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem More General Theorem Definition A real sequence a ( n ) has appropriate growth for a formal power series F ( t ) := � ∞ i =0 c i t i if a ( n + j ) ∼ a ( n ) E ( n ) j F ( δ ( n ) j ) ( n → + ∞ ) for each j with positive sequences { E ( n ) } and { δ ( n ) } → 0 . Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem More General Theorem Definition A real sequence a ( n ) has appropriate growth for a formal power series F ( t ) := � ∞ i =0 c i t i if a ( n + j ) ∼ a ( n ) E ( n ) j F ( δ ( n ) j ) ( n → + ∞ ) for each j with positive sequences { E ( n ) } and { δ ( n ) } → 0 . Question In the Hermite case we have F ( t ) := e − t 2 . E ( n ) := e A ( n ) and Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem More General Theorem Definition A real sequence a ( n ) has appropriate growth for a formal power series F ( t ) := � ∞ i =0 c i t i if a ( n + j ) ∼ a ( n ) E ( n ) j F ( δ ( n ) j ) ( n → + ∞ ) for each j with positive sequences { E ( n ) } and { δ ( n ) } → 0 . Question In the Hermite case we have F ( t ) := e − t 2 . E ( n ) := e A ( n ) and How does the shape of F ( t ) impact “limiting polynomials”? Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem More General Theorem Most General Theorem (Griffin, O, Rolen, Zagier) If a ( n ) has appropriate growth for the power series � ∞ c i t i , F ( t ) = i =0 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem More General Theorem Most General Theorem (Griffin, O, Rolen, Zagier) If a ( n ) has appropriate growth for the power series � ∞ c i t i , F ( t ) = i =0 then for each degree d ≥ 1 we have � δ ( n ) X − 1 � d � ( − 1) d − k c d − k · X k 1 a ( n ) · δ ( n ) d · J d,n lim = d ! k ! . a E ( n ) n → + ∞ k =0 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials

Hyperbolicity of Jensen polynomials Most General Theorem Some Remarks Remark (Limit Polynomials) If a : N �→ R is appropriate for F ( t ) , then � ∞ H d ( X ) · t d F ( − t ) · e Xt = � d ! . d =0 Ken Ono (U of Virginia) Hyperbolicity of Jensen polynomials