Generating relations and multivariable Aleph-function

Analysis ◽  
2018 ◽  
Vol 38 (3) ◽  
pp. 137-143 ◽  
Author(s):  
Frédéric Ayant ◽  
Dinesh Kumar
Keyword(s):  
Generating Function ◽  
Generating Relations ◽  
Function Of Two Variables ◽  
Theory Of Numbers

Abstract Srivastava and Panda have studied the simple and multiple generating relations concerning the multivariable H-function. The aim of this paper is to derive the various classes of simple and multiple generating relations involving the multivariable Aleph-function. The generating function is used in the theory of numbers, in physics and other fields of mathematics. We see the particular cases concerning the multivariable I-function, the Aleph-function of two variables and the I-function of two variables.

TRANSCENDENCE AND ALGEBRAIC INDEPENDENCE OF SERIES RELATED TO STERN'S SEQUENCE

2012 ◽  
Vol 08 (02) ◽  
pp. 361-376 ◽  
Author(s):  
PETER BUNDSCHUH
Keyword(s):  
Unit Disk ◽  
Generating Function ◽  
Algebraic Independence ◽  
Function Of Two Variables ◽  
Stern Sequence

In this same journal, Coons published recently a paper [The transcendence of series related to Stern's diatomic sequence, Int. J. Number Theory6 (2010) 211–217] on the function theoretical transcendence of the generating function of the Stern sequence, and the transcendence over ℚ of the function values at all non-zero algebraic points of the unit disk. The main aim of our paper is to prove the algebraic independence over ℚ of the values of this function and all its derivatives at the same points. The basic analytic ingredient of the proof is the hypertranscendence of the function to be shown before. Another main result concerns the generating function of the Stern polynomials. Whereas the function theoretical transcendence of this function of two variables was already shown by Coons, we prove that, for every pair of non-zero algebraic points in the unit disk, the function value either vanishes or is transcendental.

Partition theory and thermodynamics of multi-dimensional oscillator assemblies

1951 ◽  
Vol 47 (3) ◽  
pp. 591-601 ◽  
Author(s):  
V. S. Nanda
Keyword(s):  
Generating Function ◽  
Statistical Thermodynamics ◽  
Analytic Theory ◽  
Close Similarity ◽  
Thermodynamic Approach ◽  
Large Integer ◽  
Partition Problem ◽  
Partition Theory ◽  
Theory Of Numbers

The close similarity between the basic problems in statistical thermodynamics and the partition theory of numbers is now well recognized. In either case one is concerned with partitioning a large integer, under certain restrictions, which in effect means that the ‘Zustandsumme’ of a thermodynamic assembly is identical with the generating function of partitions appropriate to that assembly. The thermodynamic approach to the partition problem is of considerable interest as it has led to generalizations which so far have not yielded to the methods of the analytic theory of numbers. An interesting example is provided in a recent paper of Agarwala and Auluck (1) where the Hardy Ramanujan formula for partitions into integral powers of integers is shown to be valid for non-integral powers as well.

scholarly journals THE GENERALIZED Q-BESSEL MATRIX FUNCTION OF TWO VARIABLES

2020 ◽  
Vol 1 (4) ◽  
Author(s):  
Fadhl S. N. Alsarahi
Keyword(s):  
Difference Equation ◽  
Bessel Function ◽  
Generating Function ◽  
Matrix Function ◽  
Recurrence Relations ◽  
Special Function ◽  
Applied Mathematics ◽  
Function Of Two Variables

The Bessel function is probably the best known special function, within pure and applied mathematics. In this paper, we introduce the generalized q-analogue Bessel matrix function of two variables. Some properties of this function, such as generating function, q-difference equation, and recurrence relations are obtained.

scholarly journals Bilinear generating relations for a family of q-polynomials and generalized basic hypergeometric functions

2012 ◽  
Vol 16 (2) ◽  
pp. 191-199
Author(s):  
S. D. Purohit ◽  
V. K. Vyas ◽  
R. K. Yadav
Keyword(s):  
Generating Function ◽  
General Class ◽  
Hypergeometric Functions ◽  
Hypergeometric Polynomials ◽  
Basic Hypergeometric Functions ◽  
Generating Relations ◽  
Basic Analogue ◽  
Fox’S H Function

In this paper, we derive a bilinear q-generating function involving basic analogue of Fox's H-function and a general class of q-hypergeometric polynomials. Applications of the main results are also illustrated.

scholarly journals On Some New Linear Generating Relations Involving I-Function of Two Variables

2017 ◽  
Vol 13 (01) ◽  
pp. 23-25
Author(s):  
Raghunayak Mishra ◽  
Dr. S. S. Srivastava
Keyword(s):  
Generating Relations ◽  
Function Of Two Variables

9. Partitions

2016 ◽  
pp. 140-150
Author(s):  
Robin Wilson
Keyword(s):  
Generating Function ◽  
Circle Method ◽  
Exact Formula ◽  
New Method ◽  
Interesting Problem ◽  
Partition Problem ◽  
Number Formula ◽  
The Difference ◽  
Theory Of Numbers ◽  
The Way

How many ways can a number be split into two, three, or more pieces? ‘Partitions’ considers this interesting problem and the way in which Leonard Euler started to investigate them around 1740. Euler considered the generating function of the sequence of partition numbers and devised his pentagonal number formula. His publication Introduction to the Analysis of Infinities in 1748 outlined the difference between distinct and odd partitions. Many mathematicians worked on the partition problem, but it was not resolved until G. H. Hardy and his collaborator Srinivasa Ramanujan in 1918 published an exact formula for partition numbers using a new method in the theory of numbers called the ‘circle method’.

Generating functions involving the incomplete H-functions

Analysis ◽  
2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Kamlesh Jangid ◽  
Sunil Dutt Purohit ◽  
Kottakkaran Sooppy Nisar ◽  
Serkan Araci
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Function Theory ◽  
Generating Relations ◽  
Natural Type

Abstract In this article, for the incomplete H-functions, we obtain a set of new generating functions. The bilateral along with linear generating relations are derived for the incomplete H-functions. Many of the generating functions readily accessible in the literature are often deemed as implementations of the main findings. All the derived findings are of a natural type and can produce a variety of new results in generating function theory.

scholarly journals Some generating-function equivalences

1975 ◽  
Vol 16 (1) ◽  
pp. 34-39 ◽  
Author(s):  
H. M. Srivastava
Keyword(s):  
Orthogonal Polynomials ◽  
Generating Function ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Classical Orthogonal Polynomials ◽  
Generating Relations ◽  
Sequence Of Functions

A generalization is given of a theorem of F. Brafman [1] on the equivalence of generating relations for a certain sequence of functions. The main result, contained in Theorem 2 below, may be applied to several special functions including the classical orthogonal polynomials such as Hermite, Jacobi (and, of course, Legendre and ultraspherical), and Laguerre polynomials.

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