The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanis... more The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.
In the 1970s, Nicolas proved that the partition function p(n) is log-concave for $$ n > 25$$ n... more In the 1970s, Nicolas proved that the partition function p(n) is log-concave for $$ n > 25$$ n > 25 . In Heim et al. (Ann Comb 27(1):87–108, 2023), a precise conjecture on the log-concavity for the plane partition function $${{\textrm{pp}}}(n)$$ pp ( n ) for $$n >11$$ n > 11 was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences $$\{g_d(n)\}_{d,n}$$ { g d ( n ) } d , n with $$g_d(1)=1$$ g d ( 1 ) = 1 and $$\begin{aligned} 0 \le g_{d}\left( n\right) -n^{d}\le g_{1}\left( n\right) \left( n-1\right) ^{d-1}, \end{aligned}$$ 0 ≤ g d n - n d ≤ g 1 n n - 1 d - 1 , polynomials $$\{P_n^{g_d}(x)\}_{d,n}$$ { P n g d ( x ) } d , n given by $$\begin{aligned} \sum _{n=0}^{\infty } P_n^{g_d}(x) \, q^n := {{\textrm{exp}}}\left( x \sum _{n=1}^{\infty } g_d(n) \frac{q^n}{n} \right) =\prod _{n=1}^{\infty } \left( 1 - q^n \right) ^{-x f_d(n)}. \end{aligned}$$ ∑ n = 0 ∞ P n g d ( x ) q n : = exp x ∑ n = ...
Let g and h be real-valued arithmetic functions, positive and normalized. Specific choices within... more Let g and h be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomials P g,h n (x) := x h(n) n k=1 g(k) P g,h n−k (x), with initial value P g,h 0 (x) = 1 encode information about several classical, widely studied polynomials. This includes Chebyshev polynomials of the second kind, associated Laguerre polynomials, and the Nekrasov-Okounkov polynomials. In this paper we prove that for g = id and fixed h we obtain orthogonal polynomial sequences for positive definite functionals. Let h(n) = n s with 0 ≤ s ≤ 1. Then the sequence satisfies Turán inequalities for x ≥ 0.
We study arithmetic and asymptotic properties of polynomials provided by Q n (x) := x n k=1 k Q n... more We study arithmetic and asymptotic properties of polynomials provided by Q n (x) := x n k=1 k Q n−k (x) with initial value Q 0 (x) = 1. The coefficients satisfy a central limit theorem and a local limit theorem involving Fibonacci numbers. We apply methods of Berry and Esseen, Harper, Bender, and Canfield.
Preface.- On formal series of Jacobi forms and Borcherds products.- Finite or Infinite Number of ... more Preface.- On formal series of Jacobi forms and Borcherds products.- Finite or Infinite Number of Solutions of Polynomial Congruences in two Positive Integer Variables.- On the genus version of the basis problem II: The case of oldforms.- Restrictions of Jacobi Forms of Several Variables with special emphasis on Quaternionic Jacobi Forms.- Additive and multiplicative lifting properties of the Igusa modular form.- On the special values of certain L-series related to half-integral weight modular forms.- On explicit dimension formulas for spaces of Siegel cusp forms of degree two and their applications.- Borcherds lift on the paramodular group of level 3.- On the proportion of quadratic twists for non-vanishing and vanishing central values of L-functions attached to primitive forms.- On the Existence of Hermitian Self-Dual Extended Abelian Group Codes.- On p-adic properties of Siegel modular forms.- Bessel periods of theta lifts to GSp(1,1) and central values of some L-functions of convolution type.- Intersections of two walls of the Gottschling fundamental domain of the Siegel modular group of genus two.- Non-vanishing of L-functions associated to cusp forms of half-integral weight.- The Restriction of Octonionic Theta Constants to the Quaternionic Half-space.- Hermitian modular forms of degree 2 over the Eisenstein integers.
We study the zero distribution of non-orthogonal polynomials attached to g(n)=s(n)=n^2: Q_n^g(x)=... more We study the zero distribution of non-orthogonal polynomials attached to g(n)=s(n)=n^2: Q_n^g(x)= x ∑_k=1^n g(k) Q_n-k^g(x), Q_0^g(x):=1. It is known that the case g=id involves Chebyshev polynomials of the second kind. The zeros of Q_n^s(x) are real, simple, and are located in (-6√(3),0]. Let N_n(a,b) be the number of zeros between -6 √(3)≤ a < b ≤ 0. Then we determine a density function v(x), such that lim_n →∞N_n(a,b)/n = ∫_a^b v(x) dx. The polynomials Q_n^s(x) satisfy a four-term recursion. We present in detail an analysis of the fundamental roots and give an answer to an open question on recent work by Adams and Tran–Zumba. We extend a method proposed by Freud for orthogonal polynomials to more general systems of polynomials. We determine the underlying moments and density function for the zero distribution.
We attach to normalized (non-vanishing) arithmetic functions g and h recursively defined polynomi... more We attach to normalized (non-vanishing) arithmetic functions g and h recursively defined polynomials. Let P_0^g,h(x):=1. Then P_n^g,h(x) := x/h(n)∑_k=1^n g(k) P_n-k^g,h(x). For special g and h, we obtain the D'Arcais polynomials, which are equal to the coefficients of the -zth powers of the Dedekind η-function and are also given by Nekrasov and Okounkov as a hook length formula. Examples are offered by Pochhammer polynomials, Chebyshev polynomials of the second kind, and associated Laguerre polynomials. We present explicit formulas and identities for the coefficients of P_n^g,h(x) which separate the impact of g and h. Finally, we provide several applications.
In this paper we investigate growth properties and the zero distribution of polynomials attached ... more In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and 0<h(n) ≤ h(n+1). We put P_0^g,h(x)=1 and P_n^g,h(x) := x/h(n)∑_k=1^n g(k) P_n-k^g,h(x). As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind η-function. Here, g is the sum of divisors and h the identity function. Kostant's result on the representation of simple complex Lie algebras and Han's results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein's j-invariant, and Chebyshev polynomials of the second kind.
In this paper the image of the Saito-Kurokawa lift of level N with Dirichlet character is studied... more In this paper the image of the Saito-Kurokawa lift of level N with Dirichlet character is studied. We give a new characterization of this so called Maass Spezialschar of level N by symmetries involving Hecke operators related to Γ_0(N). We finally obtain for all prime numbers p local Maass relations. This generalizes known results for level N=1.
In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties... more In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalization by several authors have been given; on partitions with rank in a given residue class by Hou–Jagadeesan and Males, on k-regular partitions by Beckwith–Bessenrodt, on k-colored partitions by Chern, Fu, Tang, and Heim–Neuhauser on their polynomization, and Dawsey–Masri on the Andrews spt-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi–Gagola–Munagi. We offer in this paper a new proof of the Bessenrodt–Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result of Chern–Fu–Tang and its polynomization. Finally, we also obtain a new result.
Let q be an integral quadratic form of signature (2,m+2). We will show that the Siegel theta func... more Let q be an integral quadratic form of signature (2,m+2). We will show that the Siegel theta functions attached to q satisfies certain symmetries. As an application, we prove the symmetries for automorphic forms on the orthogonal group of q closely related to Heegener divisors (Borcherds lifts and automorphic Green functions).
In this paper we describe a characterization for the Maass space associated with the paramodular ... more In this paper we describe a characterization for the Maass space associated with the paramodular group of degree 2 and squarefree level N. As an application we show that the Maass space is invariant under all Hecke operators. As a consequence we conclude that the associated Siegel-Eisenstein series belongs to the Maass space.
The object of this work is the spinor L-function of degree 3 and certain degeneration related to ... more The object of this work is the spinor L-function of degree 3 and certain degeneration related to the functoriality principle. We study liftings of automorphic forms on the pair of symplectic groups (GSp(2),GSp(4)) to GSp(6). We prove cuspidality and demonstrate the compatibility with conjectures of Andrianov, Panchishkin, Deligne and Yoshida. This is done on a motivic and analytic level. We discuss an underlying torus and L-group homomorphism and put our results in the context of the Langlands program.
In this paper we introduce a new subspace of Jacobi forms of higher degree via certain relations ... more In this paper we introduce a new subspace of Jacobi forms of higher degree via certain relations among Fourier coefficients. We prove that this space can also be characterized by duality properties of certain distinguished embedded Hecke operators. We then show that this space is Hecke invariant with respect to all good Hecke operators. As explicit examples we give Eisenstein series. Conversely we show the existence of forms that are not contained in this space.
Sym 2 M k 2υ is given. This leads to a basic characterization of the Spezialschar property. The r... more Sym 2 M k 2υ is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the nonvanishing of certain special values of L-functions related to the Gross-Prasad conjecture. This is illustrated by a significant example in the paper.
We show certain symmetries for Borcherds lifts on the Hilbert modular group over a real quadratic... more We show certain symmetries for Borcherds lifts on the Hilbert modular group over a real quadratic field. We give two different proofs, the one analytic and the other arithmetic. The latter proof yields an explicit description of the action of Hecke operators on Borcherds lifts.
The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanis... more The zeros of D’Arcais polynomials, also known as Nekrasov–Okounkov polynomials, dictate the vanishing of the Fourier coefficients of powers of the Dedekind eta functions. These polynomials satisfy difference equations of hereditary type with non-constant coefficients. We relate the D’Arcais polynomials to polynomials satisfying a Volterra difference equation of convolution type. We obtain results on the transfer of the location of the zeros. As an application, we obtain an identity between Chebyshev polynomials of the second kind and 1-associated Laguerre polynomials. We obtain a new version of the Lehmer conjecture and bounds for the zeros of the Hermite polynomials.
In the 1970s, Nicolas proved that the partition function p(n) is log-concave for $$ n > 25$$ n... more In the 1970s, Nicolas proved that the partition function p(n) is log-concave for $$ n > 25$$ n > 25 . In Heim et al. (Ann Comb 27(1):87–108, 2023), a precise conjecture on the log-concavity for the plane partition function $${{\textrm{pp}}}(n)$$ pp ( n ) for $$n >11$$ n > 11 was stated. This was recently proven by Ono, Pujahari, and Rolen. In this paper, we provide a general picture. We associate to double sequences $$\{g_d(n)\}_{d,n}$$ { g d ( n ) } d , n with $$g_d(1)=1$$ g d ( 1 ) = 1 and $$\begin{aligned} 0 \le g_{d}\left( n\right) -n^{d}\le g_{1}\left( n\right) \left( n-1\right) ^{d-1}, \end{aligned}$$ 0 ≤ g d n - n d ≤ g 1 n n - 1 d - 1 , polynomials $$\{P_n^{g_d}(x)\}_{d,n}$$ { P n g d ( x ) } d , n given by $$\begin{aligned} \sum _{n=0}^{\infty } P_n^{g_d}(x) \, q^n := {{\textrm{exp}}}\left( x \sum _{n=1}^{\infty } g_d(n) \frac{q^n}{n} \right) =\prod _{n=1}^{\infty } \left( 1 - q^n \right) ^{-x f_d(n)}. \end{aligned}$$ ∑ n = 0 ∞ P n g d ( x ) q n : = exp x ∑ n = ...
Let g and h be real-valued arithmetic functions, positive and normalized. Specific choices within... more Let g and h be real-valued arithmetic functions, positive and normalized. Specific choices within the following general scheme of recursively defined polynomials P g,h n (x) := x h(n) n k=1 g(k) P g,h n−k (x), with initial value P g,h 0 (x) = 1 encode information about several classical, widely studied polynomials. This includes Chebyshev polynomials of the second kind, associated Laguerre polynomials, and the Nekrasov-Okounkov polynomials. In this paper we prove that for g = id and fixed h we obtain orthogonal polynomial sequences for positive definite functionals. Let h(n) = n s with 0 ≤ s ≤ 1. Then the sequence satisfies Turán inequalities for x ≥ 0.
We study arithmetic and asymptotic properties of polynomials provided by Q n (x) := x n k=1 k Q n... more We study arithmetic and asymptotic properties of polynomials provided by Q n (x) := x n k=1 k Q n−k (x) with initial value Q 0 (x) = 1. The coefficients satisfy a central limit theorem and a local limit theorem involving Fibonacci numbers. We apply methods of Berry and Esseen, Harper, Bender, and Canfield.
Preface.- On formal series of Jacobi forms and Borcherds products.- Finite or Infinite Number of ... more Preface.- On formal series of Jacobi forms and Borcherds products.- Finite or Infinite Number of Solutions of Polynomial Congruences in two Positive Integer Variables.- On the genus version of the basis problem II: The case of oldforms.- Restrictions of Jacobi Forms of Several Variables with special emphasis on Quaternionic Jacobi Forms.- Additive and multiplicative lifting properties of the Igusa modular form.- On the special values of certain L-series related to half-integral weight modular forms.- On explicit dimension formulas for spaces of Siegel cusp forms of degree two and their applications.- Borcherds lift on the paramodular group of level 3.- On the proportion of quadratic twists for non-vanishing and vanishing central values of L-functions attached to primitive forms.- On the Existence of Hermitian Self-Dual Extended Abelian Group Codes.- On p-adic properties of Siegel modular forms.- Bessel periods of theta lifts to GSp(1,1) and central values of some L-functions of convolution type.- Intersections of two walls of the Gottschling fundamental domain of the Siegel modular group of genus two.- Non-vanishing of L-functions associated to cusp forms of half-integral weight.- The Restriction of Octonionic Theta Constants to the Quaternionic Half-space.- Hermitian modular forms of degree 2 over the Eisenstein integers.
We study the zero distribution of non-orthogonal polynomials attached to g(n)=s(n)=n^2: Q_n^g(x)=... more We study the zero distribution of non-orthogonal polynomials attached to g(n)=s(n)=n^2: Q_n^g(x)= x ∑_k=1^n g(k) Q_n-k^g(x), Q_0^g(x):=1. It is known that the case g=id involves Chebyshev polynomials of the second kind. The zeros of Q_n^s(x) are real, simple, and are located in (-6√(3),0]. Let N_n(a,b) be the number of zeros between -6 √(3)≤ a < b ≤ 0. Then we determine a density function v(x), such that lim_n →∞N_n(a,b)/n = ∫_a^b v(x) dx. The polynomials Q_n^s(x) satisfy a four-term recursion. We present in detail an analysis of the fundamental roots and give an answer to an open question on recent work by Adams and Tran–Zumba. We extend a method proposed by Freud for orthogonal polynomials to more general systems of polynomials. We determine the underlying moments and density function for the zero distribution.
We attach to normalized (non-vanishing) arithmetic functions g and h recursively defined polynomi... more We attach to normalized (non-vanishing) arithmetic functions g and h recursively defined polynomials. Let P_0^g,h(x):=1. Then P_n^g,h(x) := x/h(n)∑_k=1^n g(k) P_n-k^g,h(x). For special g and h, we obtain the D'Arcais polynomials, which are equal to the coefficients of the -zth powers of the Dedekind η-function and are also given by Nekrasov and Okounkov as a hook length formula. Examples are offered by Pochhammer polynomials, Chebyshev polynomials of the second kind, and associated Laguerre polynomials. We present explicit formulas and identities for the coefficients of P_n^g,h(x) which separate the impact of g and h. Finally, we provide several applications.
In this paper we investigate growth properties and the zero distribution of polynomials attached ... more In this paper we investigate growth properties and the zero distribution of polynomials attached to arithmetic functions g and h, where g is normalized, of moderate growth, and 0<h(n) ≤ h(n+1). We put P_0^g,h(x)=1 and P_n^g,h(x) := x/h(n)∑_k=1^n g(k) P_n-k^g,h(x). As an application we obtain the best known result on the domain of the non-vanishing of the Fourier coefficients of powers of the Dedekind η-function. Here, g is the sum of divisors and h the identity function. Kostant's result on the representation of simple complex Lie algebras and Han's results on the Nekrasov–Okounkov hook length formula are extended. The polynomials are related to reciprocals of Eisenstein series, Klein's j-invariant, and Chebyshev polynomials of the second kind.
In this paper the image of the Saito-Kurokawa lift of level N with Dirichlet character is studied... more In this paper the image of the Saito-Kurokawa lift of level N with Dirichlet character is studied. We give a new characterization of this so called Maass Spezialschar of level N by symmetries involving Hecke operators related to Γ_0(N). We finally obtain for all prime numbers p local Maass relations. This generalizes known results for level N=1.
In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties... more In 2016 Bessenrodt–Ono discovered an inequality addressing additive and multiplicative properties of the partition function. Generalization by several authors have been given; on partitions with rank in a given residue class by Hou–Jagadeesan and Males, on k-regular partitions by Beckwith–Bessenrodt, on k-colored partitions by Chern, Fu, Tang, and Heim–Neuhauser on their polynomization, and Dawsey–Masri on the Andrews spt-function. The proofs depend on non-trivial asymptotic formulas related to the circle method on one side, or a sophisticated combinatorial proof invented by Alanazi–Gagola–Munagi. We offer in this paper a new proof of the Bessenrodt–Ono inequality, which is built on a well-known recursion formula for partition numbers. We extend the proof to the result of Chern–Fu–Tang and its polynomization. Finally, we also obtain a new result.
Let q be an integral quadratic form of signature (2,m+2). We will show that the Siegel theta func... more Let q be an integral quadratic form of signature (2,m+2). We will show that the Siegel theta functions attached to q satisfies certain symmetries. As an application, we prove the symmetries for automorphic forms on the orthogonal group of q closely related to Heegener divisors (Borcherds lifts and automorphic Green functions).
In this paper we describe a characterization for the Maass space associated with the paramodular ... more In this paper we describe a characterization for the Maass space associated with the paramodular group of degree 2 and squarefree level N. As an application we show that the Maass space is invariant under all Hecke operators. As a consequence we conclude that the associated Siegel-Eisenstein series belongs to the Maass space.
The object of this work is the spinor L-function of degree 3 and certain degeneration related to ... more The object of this work is the spinor L-function of degree 3 and certain degeneration related to the functoriality principle. We study liftings of automorphic forms on the pair of symplectic groups (GSp(2),GSp(4)) to GSp(6). We prove cuspidality and demonstrate the compatibility with conjectures of Andrianov, Panchishkin, Deligne and Yoshida. This is done on a motivic and analytic level. We discuss an underlying torus and L-group homomorphism and put our results in the context of the Langlands program.
In this paper we introduce a new subspace of Jacobi forms of higher degree via certain relations ... more In this paper we introduce a new subspace of Jacobi forms of higher degree via certain relations among Fourier coefficients. We prove that this space can also be characterized by duality properties of certain distinguished embedded Hecke operators. We then show that this space is Hecke invariant with respect to all good Hecke operators. As explicit examples we give Eisenstein series. Conversely we show the existence of forms that are not contained in this space.
Sym 2 M k 2υ is given. This leads to a basic characterization of the Spezialschar property. The r... more Sym 2 M k 2υ is given. This leads to a basic characterization of the Spezialschar property. The results of this paper are directly related to the nonvanishing of certain special values of L-functions related to the Gross-Prasad conjecture. This is illustrated by a significant example in the paper.
We show certain symmetries for Borcherds lifts on the Hilbert modular group over a real quadratic... more We show certain symmetries for Borcherds lifts on the Hilbert modular group over a real quadratic field. We give two different proofs, the one analytic and the other arithmetic. The latter proof yields an explicit description of the action of Hecke operators on Borcherds lifts.
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