scholarly journals Connection Problem for the Generalized Hypergeometric Function

2021 ◽  
Vol 64 (3) ◽  
pp. 323-348
Author(s):  
Yuya Matsuhira ◽  
Hajime Nagoya
Keyword(s):  
Hypergeometric Function ◽  
Generalized Hypergeometric Function ◽  
Connection Problem

scholarly journals Some properties of generalized hypergeometric Appell polynomials

2020 ◽  
Vol 12 (1) ◽  
pp. 129-137 ◽  
Author(s):  
L. Bedratyuk ◽  
N. Luno
Keyword(s):  
Differential Operator ◽  
Power Series ◽  
Hypergeometric Function ◽  
Explicit Formula ◽  
Series Representation ◽  
Generalized Hypergeometric Function ◽  
Appell Polynomials ◽  
Pochhammer Symbol ◽  
Formal Power ◽  
Polynomial Family

Let $x^{(n)}$ denotes the Pochhammer symbol (rising factorial) defined by the formulas $x^{(0)}=1$ and $x^{(n)}=x(x+1)(x+2)\cdots (x+n-1)$ for $n\geq 1$. In this paper, we present a new real-valued Appell-type polynomial family $A_n^{(k)}(m,x)$, $n, m \in {\mathbb{N}}_0$, $k \in {\mathbb{N}},$ every member of which is expressed by mean of the generalized hypergeometric function ${}_{p} F_q \begin{bmatrix} \begin{matrix} a_1, a_2, \ldots, a_p \:\\ b_1, b_2, \ldots, b_q \end{matrix} \: \Bigg| \:z \end{bmatrix}= \sum\limits_{k=0}^{\infty} \frac{a_1^{(k)} a_2^{(k)} \ldots a_p^{(k)}}{b_1^{(k)} b_2^{(k)} \ldots b_q^{(k)}} \frac{z^k}{k!}$ as follows $$ A_n^{(k)}(m,x)= x^n{}_{k+p} F_q \begin{bmatrix} \begin{matrix} {a_1}, {a_2}, {\ldots}, {a_p}, {\displaystyle -\frac{n}{k}}, {\displaystyle -\frac{n-1}{k}}, {\ldots}, {\displaystyle-\frac{n-k+1}{k}}\:\\ {b_1}, {b_2}, {\ldots}, {b_q} \end{matrix} \: \Bigg| \: \displaystyle \frac{m}{x^k} \end{bmatrix} $$ and, at the same time, the polynomials from this family are Appell-type polynomials. The generating exponential function of this type of polynomials is firstly discovered and the proof that they are of Appell-type ones is given. We present the differential operator formal power series representation as well as an explicit formula over the standard basis, and establish a new identity for the generalized hypergeometric function. Besides, we derive the addition, the multiplication and some other formulas for this polynomial family.

scholarly journals Certain class of p-valent functions associated with the wright generalized hypergeometric function

2010 ◽  
Vol 43 (1) ◽  
Author(s):  
M. K. Aouf ◽  
A. Shamandy ◽  
A. O. Mostafa ◽  
S. M. Madian
Keyword(s):  
Hypergeometric Function ◽  
Generalized Hypergeometric Function ◽  
New Class

AbstractUsing the Wright’s generalized hypergeometric function, we introduce a new class

scholarly journals Expansions of the Solutions of the General Heun Equation Governed by Two-Term Recurrence Relations for Coefficients

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
T. A. Ishkhanyan ◽  
T. A. Shahverdyan ◽  
A. M. Ishkhanyan
Keyword(s):  
Power Series ◽  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Recurrence Relations ◽  
Power Series Expansion ◽  
Heun Equation ◽  
Generalized Hypergeometric Function ◽  
Gamma Functions ◽  
Heun Function ◽  
Finite Sum

We examine the expansions of the solutions of the general Heun equation in terms of the Gauss hypergeometric functions. We present several expansions using functions, the forms of which differ from those applied before. In general, the coefficients of the expansions obey three-term recurrence relations. However, there exist certain choices of the parameters for which the recurrence relations become two-term. The coefficients of the expansions are then explicitly expressed in terms of the gamma functions. Discussing the termination of the presented series, we show that the finite-sum solutions of the general Heun equation in terms of generally irreducible hypergeometric functions have a representation through a single generalized hypergeometric function. Consequently, the power-series expansion of the Heun function for any such case is governed by a two-term recurrence relation.

scholarly journals Inequalities for Some Integrals Involving Modified Lommel Functions of the First Kind

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Robert E. Gaunt
Keyword(s):  
Hypergeometric Function ◽  
Generalized Hypergeometric Function ◽  
Double Inequality ◽  
Lommel Functions ◽  
Modified Struve Function ◽  
Lommel Function

AbstractIn this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind $$t_{\mu ,\nu }(x)$$tμ,ν(x). In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality, involving the modified Lommel function $$t_{\mu ,\nu }(x)$$tμ,ν(x), for a generalized hypergeometric function. The inequalities obtained in this paper generalise recent bounds for integrals involving the modified Struve function of the first kind.

On an extension of the generalized Pochhammer symbol and its applications to hypergeometric functions

2016 ◽  
Vol 09 (03) ◽  
pp. 1650064 ◽  
Author(s):  
Vivek Sahai ◽  
Ashish Verma
Keyword(s):  
Hypergeometric Function ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Function ◽  
Pochhammer Symbol ◽  
Contiguous Relations

The main object of this paper is to present a generalization of the Pochhammer symbol. We present some contiguous relations of this generalized Pochhammer symbol and use it to give an extension of the generalized hypergeometric function [Formula: see text]. Finally, we present some properties and generating functions of this extended generalized hypergeometric function.

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