Addition formulas for the $_{p}F_{p}$ and $_{p+1}F_{p}$ generalized hypergeometric functions with arbitrary parameters and their Kummer- and Euler-type transformations
We obtain addition formulas for $_{p}F_{p}$ and $_{p+1}F_{p}$ generalized hypergeometric functions with general parameters. These are utilized in conjunction with integral representations of these functions to derive Kummer- and Euler-type transformations that express $_{p}F_{p}\left(x\right)$ and $_{p+1}F_p\left(x\right)$ in the form of sums of $_{p}F_{p}\left(-x\right)$ and $_{p+1}F_p\left(-x\right)$ functions, respectively.
We derive asymptotic expansions of the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed. For both functions we consider $b/a\le 1$ and $b/a\ge 1$, with special attention for the case $a\sim b$. We use a uniform method to handle all cases of these parameters.
This paper focuses on the location of the non-asymptotic zeros of Whittaker and Kummer confluent hypergeometric functions. Based on a technique by E. Hille for the analysis of solutions of some second-order ordinary differential equations, we characterize the sign of the real part of zeros of Whittaker and Kummer functions and provide estimates on the regions of the complex plane where those zeros can be located. Our main result is a correction of a previous statement by G. E. Tsvetkov whose propagation has induced mistakes in the literature. In particular, we review some results of E. B. Saff…
In a recent paper \cite{Temme:2021:AKH} new asymptotic expansions are given for the Kummer functions $M(a,b,z)$ and $U(a,b+1,z)$ for large positive values of $a$ and $b$, with $z$ fixed and special attention for the case $a\sim b$. In this paper we extend the approach and also accept large values of $z$. The new expansions are valid when at least one of the parameters $a$, $b$, or $z$ is large. We provide numerical tables to show the performance of the expansions.
We derive a convergent expansion of the generalized hypergeometric function ${}_{p-1}F_p$ in terms of the Bessel functions ${}_{0}F_1$ that holds uniformly with respect to the argument in any horizontal strip of the complex plane. We further obtain a convergent expansion of the generalized hypergeometric function ${}_{p}F_p$ in terms of the confluent hypergeometric functions ${}_{1}F_1$ that holds uniformly in any right half-plane. For both functions, we make a further step and give convergent expansions in terms of trigonometric, exponential and rational functions that hold uniformly in the s…
We derive formulas for the construction of all inequivalent Jacobian elliptic fibrations on the Kummer surface of two non-isogeneous elliptic curves from extremal rational elliptic surfaces by rational base transformations and quadratic twists. We then show that each such decomposition yields a description of the Picard-Fuchs system satisfied by the periods of the holomorphic two-form as either a tensor product of two Gauss' hypergeometric differential equations, an Appell hypergeometric system, or a GKZ differential system. As the answer must be independent of the fibration used, identities r…
Recently, Feng, Kuznetsov and Yang discovered a very general reduction formula for a sum of products of the generalized hypergeometric functions (J. Math. Anal. Appl. 443(2016), 116--122). The main goal of this note is to present a generalization of a particular case of their identity when the generalized hypergeometric function is reduced to the Kummer function 1F1. Our generalized formula contains an additional integer shift in the bottom parameter of the Kummer function. The key ingredient of the proof is a summation formula for the Clausen series 3F2(1) with two integral parameter differen…
Recently, Rathie and Kılıçman (2014) employed Kummer-type transformation for $_{2}F_{2}(a, d+1; b, d; x)$ to develop certain classes of expansions theorems for $_{2}F_{2}(x)$ hypergeometric polynomial. Our aim is to deduce Kummer-type transformation for $_{2}F_{2}(a, d+2; b, d; x)$ and utilize it to develop some new expansion theorems for the confluent hypergeometric $_{2}F_{2}(x)$ polynomial. We also obtain a well-known result given by Kim et al. (Integral Transforms Spec. Funct. 23(6); 435-444, 2012) and many other new results as particular cases of our theorems.
In this short research note we obtain double definite integral expressions for the Kapteyn type series built by Kummer's $M$ (or confluent hypergeometric ${}_1F_1$) functions. These kind of series unify in natural way the similar fashion results for Neumann-, Schlömilch- and Kapteyn-Bessel series recently established by Pogány, Süli, Baricz and Jankov Maširević.
Integral representations of hypergeometric functions proved to be a very useful tool for studying their properties. The purpose of this paper is twofold. First, we extend the known representations to arbitrary values of the parameters and show that the extended representations can be interpreted as examples of regularizations of integrals containing Meijer's $G$ function. Second, we give new applications of both, known and extended representations. These include: inverse factorial series expansion for the Gauss type function, new information about zeros of the Bessel and Kummer type functions,…
The asymptotic expansion of the Kummer function ${}_1F_1(a; b; z)$ is examined as $z\to+\infty$ on the Stokes line $\arg\,z=0$. The correct form of the subdominant algebraic contribution is obtained for non-integer $a$. Numerical results demonstrating the accuracy of the expansion are given.
In this brief note, we show how to apply Kummer's and other quadratic transformation formulas for Gauss' and generalized hypergeometric functions in order to obtain transformation and summation formulas for series with harmonic numbers that contain one or two continuous parameters. We also give a generating function of the sequence $\frac{(a)_n (1-a)_n}{(n!)^2}H_n$ as a combination of Gauss hypergeometric function and elementary functions.
In this paper, we investigate the Euler-type integral representations for the generalized hypergeometric matrix function and develop some transformations in terms of hypergeometric matrix functions. Furthermore, unit and half arguments have been provided for several particular cases.
In Slater's 1960 standard work on confluent hypergeometric functions, also called Kummer functions, a number of asymptotic expansions of these functions can be found. We summarize expansions derived from a differential equation for large values of the $a-$parameter. We show how similar expansions can be derived by using integral representations, and we observe discrepancies with Slater's expansions.
In this paper, the product of parabolic cylinder functions $D_ν(\pm z)D_{ν+μ-1}(z)$, with different parameters $μ$ and $ν$, are established in terms of Laplace and Fourier transforms of Kummer's confluent hypergeometric functions. The provided integral representations are transformed to easily yield Nicholson-type integral forms and used to derive other series expansions for products of parabolic cylinder 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.
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Laboratory of Theoretical Physics, JINR（约飞联合核子研究所理论物理实验室）；Yerevan Physics Institute（埃里安物理研究所）；National Research University Higher School of Economics（国立研究大学高等经济大学）
We give a brief account of the key properties of elliptic hypergeometric integrals -- a relatively recently discovered top class of transcendental special functions of hypergeometric type. In particular, we describe an elliptic generalization of Euler's and Selberg's beta integrals, elliptic analogue of the Euler-Gauss hypergeometric function and some multivariable elliptic hypergeometric functions on root systems. The elliptic Fourier transformation and corresponding integral Bailey lemma technique is outlined together with a connection to the star-triangle relation and Coxeter relations for …
We consider four types of representations of solutions of GKZ system: series representations, Laplace integral representations, Euler integral representations, and Residue integral representations which will be introduced in this paper. In the former half of this paper, we provide a method for constructing integration cycles for Laplace, Residue, and Euler integral representations and relate them to series representations. In the latter half, we reformulate our integral representations in terms of direct images of $D$-modules and show their equivalence.
We propose a systematic study of transformations of $A$-hypergeometric functions. Our approach is to apply changes of variables corresponding to automorphisms of toric rings, to Euler-type integral representations of $A$-hypergeometric functions. We show that all linear $A$-hypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function $F_4$ does not admit a certain kind of Euler-type integral representation.
We show that the complex hypergeometric function describing $6j$-symbols for $SL(2,\mathbb{C})$ group is a special degeneration of the $V$-function -- an elliptic analogue of the Euler-Gauss $_2F_1$ hypergeometric function. For this function, we derive mixed difference-recurrence relations as limiting forms of the elliptic hypergeometric equation and some symmetry transformations. At the intermediate steps of computations, there emerge a function describing the $6j$-symbols for the Faddeev modular double and the corresponding difference equations and symmetry transformations.
This paper deals with the evaluation of some definite Euler-type integrals in terms of the Wright hypergeometric function. We obtain a theorem on the Wright hypergeometric function and then use this theorem to evaluate some definite integrals. Further, we derive some results as applications of these evaluations. Multi-variable cases of the derived results of this paper are also briefly discussed.
We model the dynamical behavior of a three dimensional (3-D) dissipative oscillator consisting of a $m$-block whose vertical fall occurs against a spring and which can also slide horizontally on a rigid truss rotating at a known angular speed law $ω(t)$. The $z$-vertical time law is obvious, whilst its $x$-motion along the horizontal arm is ruled by a linear differential equation to be solved through the Hermite functions and the Confluent Hypergeometric Function (CHF) $_{1}F_{1}$ (Kummer). After the rotation time law $θ(t)$ has been computed, we know completely the mass motion in a cylindrica…
It is well known that rational approximation theory involves degenerate hypergeometric functions and, in particular, the Padé approximation of the exponential function is closely related to Kummer hypergeometric functions. Recently, in the context of the study of the exponential stability of the trivial solution of delay-differential equations, a new link between the degenerate hypergeometric function and the zeros distribution of the characteristic function associated with linear delay-differential equations was emphasized. Such a link allowed the characterization of a property of time-delay …
In this paper we explore special values of Gaussian hypergeometric functions in terms of products of Euler $Γ$-functions and exponential functions of linear functions of the hypergeometric parameters. They include some classical evaluations, but the main inspiration is from the contiguity method recently applied by Akihito Ebisu.
Several expansions of the solutions of the double-confluent Heun equation in terms of the Kummer confluent hypergeometric functions are presented. Three different sets of these functions are examined. Discussing the expansions without a pre-factor, it is shown that two of these functions provide expansions the coefficients of which obey three-term recurrence relations, while for the third confluent hypergeometric function the corresponding recurrence relation generally involves five-terms. The latter relation is reduced to a three-term one only in the case when the double-confluent Heun equati…
We show that there exist infinitely many nontrivial choices of parameters of the single confluent Heun equation for which the three-term recurrence relations governing the expansions of the solutions in terms of the confluent hypergeometric functions 1F1 and 0F1 are reduced to two-term ones. In such cases the expansion coefficients are explicitly calculated in terms of the Euler gamma functions.
Reduction formulas for sums of products of hypergeometric functions can be traced back to Euler. This topic has an intimate connection to summation and transformation formulas, contiguous relations and algebraic properties of the (generalized) hypergeometric differential equation. Over recent several years, important discoveries have been made in this subject by Gorelov, Ebisu, Beukers and Jouhet and Feng, Kuznetsov and Yang. In this paper, we give a generalization of Feng, Kuznetsov and Yang identity covering also Ebisu's and Gorelov's formulas as particular cases.
