On Jackson’s proof of Ramanujan’s 1ψ1 summation formula

2018 ◽  
Vol 14 (02) ◽  
pp. 313-328
Author(s):  
Jorge Luis Cimadevilla Villacorta
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Summation Formula ◽  
Ramanujan’S Formula

In this paper, the author proves some basic hypergeometric series which utilizes the same ideas that Margaret Jackson used to give a proof of Ramanujan’s [Formula: see text] summation formula.

scholarly journals Some q-Supercongruences from Transformation Formulas for Basic Hypergeometric Series

2020 ◽  
Author(s):  
Victor J. W. Guo ◽  
Michael J. Schlosser
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Double Series ◽  
Summation Formula ◽  
Transformation Formula ◽  
Quadratic Transformation ◽  
Ultraspherical Polynomials ◽  
Higher Power ◽  
Series Transformation ◽  
Transformation Formulas

AbstractSeveral new q-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the first author in collaboration with Zudilin. More concretely, the results in this paper include q-analogues of supercongruences (referring to p-adic identities remaining valid for some higher power of p) established by Long, by Long and Ramakrishna, and several other q-supercongruences. The six basic hypergeometric transformation formulas which are made use of are Watson’s transformation, a quadratic transformation of Rahman, a cubic transformation of Gasper and Rahman, a quartic transformation of Gasper and Rahman, a double series transformation of Ismail, Rahman and Suslov, and a new transformation formula for a nonterminating very-well-poised $${}_{12}\phi _{11}$$ 12 ϕ 11 series. Also, the nonterminating q-Dixon summation formula is used. A special case of the new $${}_{12}\phi _{11}$$ 12 ϕ 11 transformation formula is further utilized to obtain a generalization of Rogers’ linearization formula for the continuous q-ultraspherical polynomials.

scholarly journals On a New Summation Formula for Basic Bilateral Hypergeometric Series and Its Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-7
Author(s):  
D. D. Somashekara ◽  
K. Narasimha Murthy ◽  
S. L. Shalini
Keyword(s):  
Hypergeometric Series ◽  
Beta Function ◽  
Basic Hypergeometric Series ◽  
Summation Formula ◽  
Bilateral Basic Hypergeometric Series

We have obtained a new summation formula for bilateral basic hypergeometric series by the method of parameter augmentation and demonstrated its various uses leading to some development of etafunctions, -gamma, and -beta function identities.

Generalized basic hypergeometric series with unconnected bases

1967 ◽  
Vol 63 (3) ◽  
pp. 727-734 ◽  
Author(s):  
R. P. Agarwal ◽  
Arun Verma
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Theta Functions ◽  
Summation Formula ◽  
Mock Theta Functions ◽  
Transformation Theory ◽  
Infinite Products ◽  
Bilateral Basic Hypergeometric Series ◽  
Basic Series ◽  
Made In

In a series of recent papers Verma and Upadhyay (7,8,9) developed the theory of basic hypergeometric series with two bases q and q½. These investigations were made in an attempt to discover a summation formula for a bilateral basic hypergeometric series 2Ψ2 analogous to that for a 2H2 (cf. Bailey (2,3)) and in finding relations between certain q-infinite products. In one of their papers they mentioned that it did not seem possible to develop the corresponding general theory for basic series with two unconnected bases q and q1. A recent paper by Andrews (1) indicates that transformations between basic hypergeometric series with two unconnected bases can be very interesting and useful in the study of ‘mock’ theta functions and their extensions. Besides this interest, such a theory also enables one to extend the entire existing transformation theory of the generalized basic hypergeometric series.

scholarly journals Approach of q-Derivative Operators to Terminating q-Series Formulae

2018 ◽  
Vol 26 (2) ◽  
pp. 99-111
Author(s):  
Xiaoyuan Wang ◽  
Wenchang Chu
Keyword(s):  
Hypergeometric Series ◽  
Basic Hypergeometric Series ◽  
Operator Approach ◽  
Derivative Operator ◽  
Summation Formulae

AbstractThe q-derivative operator approach is illustrated by reviewing several typical summation formulae of terminating basic hypergeometric series.

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