Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn generalization of Baileys identity involving product of generalized hypergeometric series
281
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
The aim of this research paper is to obtain explicit expressions of (i) $ {}_1F_1 left[begin{array}{c} alpha 2alpha + i end{array} ; x right]. {}_1F_1left[ begin{array}{c} beta 2beta + j end{array} ; x right]$ (ii) ${}_1F_1 left[ begin{array}{c} alpha 2alpha - i end{array} ; x right] . {}_1F_1 left[ begin{array}{c} beta 2beta - j end{array} ; x right]$ (iii) ${}_1F_1 left[ begin{array}{c} alpha 2alpha + i end{array} ; x right] . {}_1F_1 left[begin{array}{c} beta 2beta - j end{array} ; x right]$ in the most general form for any $i,j=0,1,2,ldots$ For $i=j=0$, we recover well known and useful identity due to Bailey. The results are derived with the help of a well known Baileys formula involving products of generalized hypergeometric series and generalization of Kummers second transformation formulas available in the literature. A few interesting new as well as known special cases have also been given.
rate research
Read More
Using generalized hypergeometric functions to perform symbolic manipulation of equations is of great importance to pure and applied scientists. There are in the literature a great number of identities for the Meijer-G function. On the other hand, when more complex expressions arise, the latter function is not capable of representing them. The H-function is an alternative to overcome this issue, as it is a generalization of the Meijer-G function. In the present paper, a new identity for the H-function is derived. In short, this result enables one to split a particular H-function into the sum of two other H-functions. The new relation in addition to an old result are applied to the summation of hypergeometric series. Finally, some relations between H-functions and elementary functions are built
E661 in the Enestrom index. This was originally published as Variae considerationes circa series hypergeometricas (1776). In this paper Euler is looking at the asymptotic behavior of infinite products that are similar to the Gamma function. He looks at the relations between some infinite products and integrals. He takes the logarithm of these infinite products, and expands these using the Euler-Maclaurin summation formula. In section 14, Euler seems to be rederiving some of the results he already proved in the paper. However I do not see how these derivations are different. If any readers think they understand please I would appreciate it if you could email me. I am presently examining Eulers work on analytic number theory. The two main topics I want to understand are the analytic continuation of analytic functions and the connection to divergent series, and the asymptotic behavior of the Gamma function.
In this note, we aim to provide generalizations of (i) Knuths old sum (or Reed Dawson identity) and (ii) Riordans identity using a hypergeometric series approach.
The purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forellis theorem on the complex analyticity of the functions that are: (1) $mathcal{C}^infty$ smooth at a point, and (2) holomorphic along the complex integral curves generated by a contracting holomorphic vector field with an isolated zero at the same point.
We formulate general principles of building hypergeometric type series from the Jacobi theta functions that generalize the plain and basic hypergeometric series. Single and multivariable elliptic hypergeometric series are considered in detail. A characterization theorem for a single variable totally elliptic hypergeometric series is proved.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
