scholarly journals A Generalized Inverse Binomial Summation Theorem and Some Hypergeometric Transformation Formulas

2016 ◽  
Vol 2016 ◽  
pp. 1-14
Author(s):  
S. M. Ripon
Keyword(s):  
Generalized Inverse ◽  
Binomial Coefficient ◽  
Bell Polynomials ◽  
Binomial Theorem ◽  
Summation Theorem ◽  
Transformation Formulas

A generalized binomial theorem is developed in terms of Bell polynomials and by applying this identity some sums involving inverse binomial coefficient are calculated. A technique is derived for calculating a class of hypergeometric transformation formulas and also some curious q series identities.

scholarly journals General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1102
Author(s):  
Yashoverdhan Vyas ◽  
Hari M. Srivastava ◽  
Shivani Pathak ◽  
Kalpana Fatawat
Keyword(s):  
Hypergeometric Functions ◽  
Theory Theory ◽  
Binomial Theorem ◽  
Generalized Hypergeometric Functions ◽  
Quantum Calculus ◽  
The Third ◽  
Summation Formulas ◽  
Symmetric Quantum ◽  
Summation Theorem

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

scholarly journals Applications of the Generalized Kummer’s Summation Theorem to Transformation Formulas and Generating Functions

2018 ◽  
Vol 3 (2) ◽  
pp. 331-338 ◽  
Author(s):  
Ahmed Ali Atash ◽  
Hussein Saleh Bellehaj
Keyword(s):  
Generating Functions ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
General Transformation ◽  
Summation Theorem ◽  
Transformation Formulas

AbstractIn this paper, we establish two general transformation formulas for Exton’s quadruple hypergeometric functions K5 and K12 by application of the generalized Kummer’s summation theorem. Further, a number of generating functions for Jacobi polynomials are also derived as an applications of our main results.

Binomial Coefficient Formulas by General Reasoning

1968 ◽  
Vol 61 (4) ◽  
pp. 399-402
Author(s):  
Jack M. Elkin
Keyword(s):  
Binomial Coefficient ◽  
Algebraic Manipulation ◽  
Binomial Coefficients ◽  
Binomial Theorem ◽  
Relative Ease ◽  
Methods Of Analysis ◽  
Summation Of Series ◽  
Mathematical Formulas ◽  
High Degree ◽  
Pascal's Triangle

The binomial coefficients are an almost endless source of formulas for the summation of series. A reference to the “Problems for Solution” pages of the American Mathematical Monthly or to an advanced collection of mathematical formulas will convince anyone who has not yet discovered this for himself. Some of these series summations can be derived with relative ease with the help of the binomial theorem or Pascal's triangle; many require a high degree of virtuosity in algebraic manipulation and, often, advanced methods of analysis. A number of them can be obtained simply by reasoning logically about the meaning of certain combinatorial expressions, with recourse to only a minimum of algebra or to none at all. These, naturally, have a special appeal of their own, and it is the purpose of this article to illustrate several such derivations.

scholarly journals On the quantum differentiation of smooth real-valued functions

2017 ◽  
Author(s):  
Kolosov Petro
Keyword(s):  
Series Representation ◽  
Numerical Differentiation ◽  
Interpolation Formula ◽  
Binomial Coefficient ◽  
Discrete Analog ◽  
Binomial Theorem ◽  
Quantum Calculus ◽  
Time Scale Calculus ◽  
Power Difference ◽  
Quantum Difference

Calculating the value of $C^{k\in\{1,\infty\}}$ class of smoothness real-valued function's derivative in point of $\mathbb{R}^+$ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and $q$-difference operator. $(P,q)$-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using $q$-difference and $p,q$-power difference is shown.MSC 2010: 26A24, 05A30, 41A58arXiv:1705.02516Keywords: Quantum calculus, Quantum algebra, Power quantum calculus, Quantum difference, q-derivative, Jackson derivative, q-calculus, q-difference, Time Scale Calculus, Series Expansion, Taylor's theorem, Taylor's formula, Taylor's series, Taylor's polynomial, Analytic function, Series representation, Derivative, Differential calculus, Difference Equations, Numerical Differentiation, Polynomial, Exponential function, Exponentiation, Binomial coefficient, Binomial theorem, Binomial expansion, Mathematics, Numerical analysis, Mathematical analysis, arXiv, Preprint

scholarly journals Explicit, Determinantal, and Recurrent Formulas of Generalized Eulerian Polynomials

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 37
Author(s):  
Yan Wang ◽  
Muhammet Cihat Dağli ◽  
Xi-Min Liu ◽  
Feng Qi
Keyword(s):  
General Formula ◽  
Bell Polynomials ◽  
Eulerian Polynomials ◽  
Differentiable Functions ◽  
Faà Di Bruno Formula ◽  
Derivatives Of

In the paper, by virtue of the Faà di Bruno formula, with the aid of some properties of the Bell polynomials of the second kind, and by means of a general formula for derivatives of the ratio between two differentiable functions, the authors establish explicit, determinantal, and recurrent formulas for generalized Eulerian polynomials.

scholarly journals Some identities for degenerate complete and incomplete r-Bell polynomials

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Jongkyum Kwon ◽  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim
Keyword(s):  
Bell Polynomials
Sign in / Sign up

Export Citation Format

Share Document