The generalized binomial theorem

2012 ◽  
pp. 315-321
Author(s):  
Murali Rao
Keyword(s):  
Binomial Theorem

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.

Application of Maple on Solving the Differential Problem of Rational Functions

2013 ◽  
Vol 479-480 ◽  
pp. 855-860
Author(s):  
Chii Huei Yu
Keyword(s):  
Problem Solving ◽  
Research Methods ◽  
Research Method ◽  
Rational Functions ◽  
The Other ◽  
Mathematical Software ◽  
Differential Problem ◽  
Binomial Theorem ◽  
Other Hand ◽  
Derivatives Of

This paper uses the mathematical software Maple as the auxiliary tool to study the differential problem of four types of rational functions. We can obtain the closed forms of any order derivatives of these rational functions by using binomial theorem. On the other hand, we propose four examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. For this reason, Maple provides insights and guidance regarding problem-solving methods.

scholarly journals XV. A new demonstration of the binomial theorem, when the exponent is a positive or negative fraction. By the Rev. Abram Robertson, A. M. F. R. S. Savilian Professor of Geometry in the University of Oxford. In a letter to Davies Giddy, Esq. F. R. S

1806 ◽  
Vol 96 ◽  
pp. 305-326 ◽  
Keyword(s):  
Royal Society ◽  
Mathematical Science ◽  
Binomial Theorem ◽  
Useful Knowledge ◽  
University Of Oxford ◽  
The University ◽  
The Way

Dear Sir, Being perfectly convinced of your love of mathematical science, and your extensive acquirements in it, I submit to your perusal a new demonstration of the binomial theorem, when the exponent is a positive or negative fraction. As I am a strenuous advocate for smoothing the way to the acquisition of useful knowledge, i deem the following articles of some importance ; and unless I were equally sincere in this persuasion, and in that of your desire to promote mathemati­cal studies, in requesting the perusal, I should accuse myself of an attempt to trifle with your valuable time. The following demonstration is new only to the extent above mentioned ; but in order that the reader may perceive the proof to be complete, a successive perusal of all the articles is necessary. As far as it relates to the raising of in­tegral powers, it is in substance the same with one which I drew up in the year 1794, and which was honoured with a place in the Philosophical Transactions for 1795. If, therefore, you think the following demonstration worthy the attention of mathematicians, you will much oblige me by presenting it to the Royal Society.

The binomial theorem

1983 ◽  
pp. 45-59
Author(s):  
B.D. Bunday ◽  
H. Mulholland
Keyword(s):  
Binomial Theorem
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