The differential transformation method and Miller's recurrence

Degenerate Miller-Paris transformations

Important new transformations for the generalized hypergeometric functions with integral parameter differences have been discovered some years ago by Miller and Paris and studied in detail in a series of papers by a number of authors. These transformations fail if the free bottom parameter is greater than a free top parameter by a small positive integer. In this paper we fill this gap in the theory of Miller-Paris transformations by computing the limit cases of these transformations in such previously prohibited situations. This leads to a number of new transformation and summation formulas in…

Alternative approach to Miller-Paris transformations and their extensions

Miller-Paris transformations are extensions of Euler's transformations for the Gauss hypergeometric functions to generalized hypergeometric functions of higher-order having integral parameter differences (IPD). In our recent work we computed the degenerate versions of these transformations corresponding to the case when one parameter difference is equal to a negative integer. The purpose of this paper is to present an independent new derivation of both the general and the degenerate forms of Miller-Paris transformations. In doing so we employ the generalized Stieltjes transform representation …

Extensions of Karlsson-Minton summation theorem and some consequences of the first Miller-Paris transformation

In this paper we give several independent extensions of the Karlsson-Minton summation formula for the generalized hypergeometric function with integral parameters differences. In particular, we examine the "prohibited" values for the integer top parameter in Minton's formula, extend one unit negative difference in Karlsson's formula to a finite number of integer negative differences and establish known and new summation and transformation formulas when the unit negative difference is allowed to take arbitrary values. We also present a recurrence relation reducing the case of integer negative d…

与受扰动的 $R_I$ 型递推关系相关的谱变换

Spectral transformation associated with a perturbed $R_I$ type recurrence relation

本文研究受扰动 $R_I$ 型递推关系下正交多项式的结构与谱变换，证明传递矩阵法更高效，并分析共递推、共膨胀及零点性质。

例外 Laurent 双正交多项式的递推关系

Recurrence relations of Exceptional Laurent biorthogonal polynomials

本文通过 Darboux 变换构造例外 Laurent 双正交多项式，并给出其具有 $3l_0+4$ 项的递推关系，证明中利用后向算子将例外多项式映射为经典多项式。

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。