scholarly journals A new transformation formula involving derived WP-bailey pair and its applications

Filomat ◽  
2020 ◽  
Vol 34 (13) ◽  
pp. 4245-4252
Author(s):  
Zhizheng Zhang ◽  
Jing Gu ◽  
Hanfei Song
Keyword(s):  
Transformation Formula ◽  
Lambert Series ◽  
Summation Formulas ◽  
Transformation Formulas

The main purpose of this paper is to obtain a new transformation formula involving the derived WP-Bailey pair. As applications, by using two 10?9 summation formulas, some transformation formulas in terms of generalized Lambert series are obtained.

scholarly journals An identity involving Nörlund polynomials

1991 ◽  
Vol 43 (2) ◽  
pp. 307-315
Author(s):  
A.E. Özlük ◽  
C. Snyder
Keyword(s):  
Lattice Points ◽  
Transformation Formula ◽  
Lambert Series

We prove an identity involving Nörlund polynomials, the proof of which is elementary and involves the enumeration of lattice points. The identity is slightly stronger than an identity of Carlitz which he obtained by using Apostol's transformation formula for Lambert series.

scholarly journals Transformation formulas for terminating Saalschützian hypergeometric series of unit argument

1995 ◽  
Vol 8 (2) ◽  
pp. 189-194
Author(s):  
Wolfgang Bühring
Keyword(s):  
Recurrence Relation ◽  
Hypergeometric Series ◽  
Summation Formula ◽  
Summation Formulas ◽  
Finite Sums ◽  
Transformation Formulas ◽  
Independent Parameters

Transformation formulas for terminating Saalschützian hypergeometric series of unit argument p+1Fp(1) are presented. They generalize the Saalschützian summation formula for 3F2(1). Formulas for p=3,4,5 are obtained explicitly, and a recurrence relation is proved by means of which the corresponding formulas can also be derived for larger p. The Gaussian summation formula can be derived from the Saalschützian formula by a limiting process, and the same is true for the corresponding generalized formulas. By comparison with generalized Gaussian summation formulas obtained earlier in a different way, two identities for finite sums involving terminating 3F2(1) series are found. They depend on four or six independent parameters, respectively.

scholarly journals Corollaries and multiple extensions of Gessel and Stanton hypergeometric summation formulas

2021 ◽  
Vol 25 (1) ◽  
pp. 21-31
Author(s):  
Thomas Ernst ◽  
Per W. Karlsson
Keyword(s):  
Special Function ◽  
Summation Formulas ◽  
Multiple Transformation ◽  
Euler Transformation ◽  
Transformation Formulas ◽  
Parameter Values ◽  
Appell Functions

We find some new simple hypergeometric formulas in the footsteps of the important article by Gessel and Stanton. These are multiple reduction formulas, multiple summation formulas, as well as multiple transformation formulas for special Kampé de Fériet functions and Appell functions. The hypergeometric summation formulas have special function arguments in Q and parameter values in N or C. The proofs use Pfaff-Kummer transformation, Euler transformation, or an improved form of Slater reversion.

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 Transformation of Some Lambert Series and Cotangent Sums

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 840
Author(s):  
Namhoon Kim
Keyword(s):  
Half Plane ◽  
Modular Group ◽  
Transformation Formula ◽  
Contour Integral ◽  
Dedekind Sums ◽  
Lambert Series ◽  
Simple Derivation ◽  
Special Cases ◽  
Upper Half Plane

By considering a contour integral of a cotangent sum, we give a simple derivation of a transformation formula of the series A ( τ , s ) = ∑ n = 1 ∞ σ s − 1 ( n ) e 2 π i n τ for complex s under the action of the modular group on τ in the upper half plane. Some special cases directly give expressions of generalized Dedekind sums as cotangent sums.

PROBABILISTIC DERIVATION OF A q-POLYNOMIALS TRANSFORMATION FORMULA

2013 ◽  
Vol 16 (02) ◽  
pp. 1350017 ◽  
Author(s):  
MINGJIN WANG
Keyword(s):  
Convergence Theorem ◽  
Summation Formula ◽  
Transformation Formula ◽  
Dominated Convergence ◽  
Transformation Formulas

In this paper, we use the distribution W(x; q) and the Lebesgue's dominated convergence theorem to give a probabilistic derivation of a transformation formula for the Al-Salam-Carlitz polynomials. Some applications of the transformation are also given, which include an extension of the Heine's 2ϕ1 transformation formulas and an extension of the Bailey-Daum summation formula.

scholarly journals Some new transformation formulas deriving from Bailey pairs and WP-Bailey pairs

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 941-953
Author(s):  
Shuyuan Nie ◽  
Zhizheng Zhang
Keyword(s):  
Hypergeometric Series ◽  
Bailey Pairs ◽  
Summation Formulas ◽  
Transformation Formulas

The purpose of this paper is to derive a new Bailey pair and three new WP-Bailey pairs from four summation formulas of the multibasic hypergeometric series. As applications, we will use them to obtain many new transformation formulas for basic and multibasic hypergeometric series.

scholarly journals A Note on the New Extended Beta Function with Its Application

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Vandana Palsaniya ◽  
Ekta Mittal ◽  
Sunil Joshi ◽  
D. L. Suthar
Keyword(s):  
Hypergeometric Function ◽  
Beta Function ◽  
Moment Generating Function ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Summation Formulas ◽  
Derivative Formulas ◽  
New Type ◽  
Extended Beta Function ◽  
Transformation Formulas

The purpose of this research is to provide a systematic review of a new type of extended beta function and hypergeometric function using a confluent hypergeometric function, as well as to examine various belongings and formulas of the new type of extended beta function, such as integral representations, derivative formulas, transformation formulas, and summation formulas. In addition, we also investigate extended Riemann–Liouville (R-L) fractional integral operator with associated properties. Furthermore, we develop new beta distribution and present graphically the relation between moment generating function and ℓ .

scholarly journals A study on certain properties of generalized special functions defined by Fox-Wright function

2020 ◽  
Vol 5 (1) ◽  
pp. 147-162
Author(s):  
Enes Ata ◽  
İ. Onur Kıymaz
Keyword(s):  
Hypergeometric Function ◽  
Special Functions ◽  
Confluent Hypergeometric Function ◽  
Integral Representations ◽  
Gauss Hypergeometric Function ◽  
Wright Function ◽  
Summation Formulas ◽  
Frequent Use ◽  
Derivative Formulas ◽  
Transformation Formulas

AbstractIn this study, motivated by the frequent use of Fox-Wright function in the theory of special functions, we first introduced new generalizations of gamma and beta functions with the help of Fox-Wright function. Then by using these functions, we defined generalized Gauss hypergeometric function and generalized confluent hypergeometric function. For all the generalized functions we have defined, we obtained their integral representations, summation formulas, transformation formulas, derivative formulas and difference formulas. Also, we calculated the Mellin transformations of these functions.

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