Padé interpolation and hypergeometric series

Elliptic Hypergeometric Summations by Taylor Series Expansion and Interpolation

We use elliptic Taylor series expansions and interpolation to deduce a number of summations for elliptic hypergeometric series. We extend to the well-poised elliptic case results that in the $q$-case have previously been obtained by Cooper and by Ismail and Stanton. We also provide identities involving S. Bhargava's cubic theta functions.

Convergent Interpolation to Cauchy Integrals over Analytic Arcs

We consider multipoint Padé approximation to Cauchy transforms of complex measures. We show that if the support of a measure is an analytic Jordan arc and if the measure itself is absolutely continuous with respect to the equilibrium distribution of that arc with Dini-smooth non-vanishing density, then the diagonal multipoint Padé approximants associated with appropriate interpolation schemes converge locally uniformly to the approximated Cauchy transform in the complement of the arc. This asymptotic behavior of Padé approximants is deduced from the analysis of underlying non-Hermitian orthogo…

Two multivariate quadratic transformations of elliptic hypergeometric integrals

Eric Rains conjectured several quadratic transformations between multivariate elliptic hypergeometric functions in "Elliptic Littlewood Identities", with the integrand multiplied by interpolation functions. In this article two of these conjectures are proven in the case where the interpolation functions are constant, and we obtain a third conjecture as a corollary. Two other equations for elliptic Selberg integrals with 10 parameters, two of which multiplying to pq/t, are given, as they are needed in the proof. The proofs consist essentially of a calculation which strings together many ellipti…

Hermite-Padé逼近、多重正交多项式与多维Toda方程

Hermite-Padé approximation, multiple orthogonal polynomials, and multidimensional Toda equations

本文综述了Hermite-Padé逼近、多重正交多项式与多维Toda方程之间的联系，并介绍了向插值问题和非交换代数的推广。

Interpolation by generalized exponential sums with equal weights

Here we solve Padé and Prony interpolation problems for the generalized exponential sums with equal weights: $$H_n(z; h)=\fracμ{n}\sum_{k=1}^n h(λ_k z),\quad \text{where}\quad μ,λ_k\in \mathbb{C},$$ and $h$ is a fixed analytic function under few natural assumptions. The interpolation of a function $f$ by $H_n$ is due to properly chosen $μ$ and $\{λ_k\}_{k=1}^n$, which depend on $f$, $h$ and $n$. The sums $H_n$ are related to the $h$-sums and generalized exponential sums, i.e. to $$\mathcal{H}^*_n(z; h)=\sum_{k=1}^n λ_k h(λ_k z)\quad \text{and}\quad\mathcal{H}_n(z; h):=\sum_{k=1}^n μ_k h(λ_k z)…

Euler and Laplace integral representations of GKZ hypergeometric functions

We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. We establish a combinatorial method of constructing a basis of the rapid decay homology group associated to Euler-Laplace integral with a nice intersection property. This construction yields a remarkable expansion formula of cohomology intersection numbers in terms of GKZ hypergeometric series. As an application, we obtain closed formulas of the quadratic relations of Aomoto-Gelfand hypergeometric functions and their confluent analogue in terms of bipartite graphs.

A quadratic formula for basic hypergeometric series related to Askey-Wilson polynomials

We prove a general quadratic formula for basic hypergeometric series, from which simple proofs of several recent determinant and Pfaffian formulas are obtained. A special case of the quadratic formula is actually related to a Gram determinant formula for Askey-Wilson polynomials. We also show how to derive a recent double-sum formula for the moments of Askey-Wilson polynomials from Newton's interpolation formula.

Branching rules for symmetric Macdonald polynomials and sl_n basic hypergeometric series

A one-parameter generalisation R_λ(X;b) of the symmetric Macdonald polynomials and interpolations Macdonald polynomials is studied from the point of view of branching rules. We establish a Pieri formula, evaluation symmetry, principal specialisation formula and q-difference equation for R_λ(X;b). We also prove a new multiple q-Gauss summation formula and several further results for sl_n basic hypergeometric series based on R_λ(X;b).

Ramanujan's Master theorem for the hypergeometric Fourier transform on root systems

Ramanujan's Master theorem states that, under suitable conditions, the Mellin transform of an alternating power series provides an interpolation formula for the coefficients of this series. Ramanujan applied this theorem to compute several definite integrals and power series, which explains why it is referred to as the "Master Theorem". In this paper we prove an analogue of Ramanujan's Master theorem for the hypergeometric Fourier transform on root systems. This theorem generalizes to arbitrary positive multiplicity functions the results previously proven by the same authors for the spherical …

The linear pencil approach to rational interpolation

It is possible to generalize the fruitful interaction between (real or complex) Jacobi matrices, orthogonal polynomials and Pade approximants at infinity by considering rational interpolants, (bi-)orthogonal rational functions and linear pencils zB-A of two tridiagonal matrices A, B, following Spiridonov and Zhedanov. In the present paper, beside revisiting the underlying generalized Favard theorem, we suggest a new criterion for the resolvent set of this linear pencil in terms of the underlying associated rational functions. This enables us to generalize several convergence results for Pade a…

Chowla-Selberg 时期线性独立性度量

Linear independence measures for Chowla--Selberg periods

通过同时Padé近似估计超几何函数，研究特定d和D下1、π√d、Ω_D/π和π/Ω_D的线性形式，揭示其线性独立性。

欧拉恒等式与克劳森恒等式的多项式扰动

Polynomial perturbations of Euler's and Clausen's identities

针对两个超几何级数乘积通常非超几何的问题，本文通过多项式扰动扩展欧拉变换与克劳森恒等式，提出Miller-Paris算子并给出简化证明，实现超几何级数乘积向单个超几何级数的转化。

Approximation by amplitude and frequency operators

We study Padé interpolation at the node $z=0$ of functions $f(z)=\sum_{m=0}^{\infty} f_m z^m$, analytic in a neighbourhood of this node, by amplitude and frequency operators (sums) of the form $$ \sum_{k=1}^n μ_k h(λ_k z), \qquad μ_k,λ_k\in \mathbb{C}. $$ Here $h(z)=\sum_{m=0}^{\infty} h_m z^m$, $h_m\ne 0$, is a fixed (basis) function, analytic at the origin, and the interpolation is carried out by an appropriate choice of amplitudes $μ_k $ and frequencies $λ_k$. The solvability of the $2n$-multiple interpolation problem is determined by the solvability of the associated moment problem $$ \sum…