By making use of some techniques based upon certain inverse new pairs of symbolic operators, the author investigate several decomposition formulas associated with Lauricella's hypergeometric functions $F_A^{(r)}, F_B^{(r)}, F_C^{(r)}$ and $F_D^{(r)}$ in $r$ variables. In the three-variable case some of these operational representations are constructed and applied in order to derive the corresponding decomposition formulas when $r = 3$ . With the help of these new inverse pairs of symbolic operators, a total 20 decomposition formulas and integral representations are found.
We give an example of solutions of the connection problem associated with a certain system of linear $q$-difference equations recently introduced by Park. The result contains a connection formulas of the $q$-Lauricella hypergeometric function $φ_{D}$ and those of the $q$-generalized hypergeometric function ${}_{N+1}φ_{N}$ as special cases.
We present a new methodology, suitable for implementation on computer, to perform the $ε$-expansion of hypergeometric functions with linear $ε$ dependent Pochhammer parameters in any number of variables. Our approach allows one to perform Taylor as well as Laurent series expansion of multivariable hypergeometric functions. Each of the coefficients of $ε$ in the series expansion is expressed as a linear combination of multivariable hypergeometric functions with the same domain of convergence as that of the original hypergeometric function. We present illustrative examples of hypergeometric func…
HYPERgeometric functions DIfferential REduction: Mathematica-based packages for the differential reduction of generalizedhypergeometric functions: Fc hypergeometric function of three variables
Analytic continuations and numerical evaluation of the Appell $F_1$, $F_3$, Lauricella $F_D^{(3)}$ and Lauricella-Saran $F_S^{(3)}$ and their Application to Feynman Integrals
HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: $F_D$ and $F_S$ Horn-type hypergeometric functions of three variables
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 …
Hypergeometric structures in single and multiscale Feynman integrals emerge in a wide class of topologies. Using integration-by-parts relations, associated master or scalar integrals have to be calculated. For this purpose it appears useful to devise an automated method which recognizes the respective (partial) differential equations related to the corresponding higher transcendental functions. We solve these equations through associated recursions of the expansion coefficient of the multivalued formal Taylor series. The expansion coefficients can be determined using either the package {\tt Si…
Wilson polynomials/functions and intertwining operators for the generic quantum superintegrable system on the 2-sphere
自2007年以来，人们已经知道威尔逊和拉卡多项式可以被表征为2-球面上的量子超可积系统的二次对称代数的不可约表示的基函数，$HΨ=EΨ$，具有一般的3参数势。显然，多项式是$H$的对称算子$L_1$的一个本征基关于另一个对称算子$L_2$的本征基的展开系数，但确切的关系似乎没有明确表示。我们工作的细节展开显示，明确地，如何多项式出现，以及如何这些功能的主要属性：措施，3项递归关系，二阶差分方程，这些关系的对偶性，排列对称性，交织运营商和威尔逊函数的替代推导-遵循这个量子系统的对称性。具有一般势的$n$ -球面上的多变量Wilson多项式与量子超可积系统之间的关系一直是人们关注的热点，这些结果有助于推广.将这个二次代数的不可约表示的函数空间实现收缩到其它超可积系统上，可以得到正交超几何多项式的完整Askey方案。所有这些超可积系统的压缩都是由$so(3, C )$和$e(2, C)$的Wigner李代数压缩唯一诱导的.所有产生的多项式都可以解释为量子展开系数。重要的是将这个过程扩展到更高的维度。
