scholarly journals Proofs of some conjectures on monotonicity of ratios of Kummer, Gauss and generalized hypergeometric functions

Analysis ◽  
2016 ◽  
Vol 36 (4) ◽  
Author(s):  
Sergei M. Sitnik ◽  
Khaled Mehrez
Keyword(s):  
Hypergeometric Functions ◽  
Exponential Series ◽  
Generalized Hypergeometric Functions ◽  
General Conjecture ◽  
Turán Type Inequalities

AbstractIn 1993 one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. These lead to a more general conjecture on monotonicity of ratios of Kummer hypergeometric functions, which remained open ever since. In this paper we prove some conjectures for Kummer hypergeometric functions and its further generalizations for Gauss and generalized hypergeometric functions. The results are also closely connected with Turán-type inequalities.

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.

Clebsh–Gordan coefficients for the algebra 𝔤𝔩₃ and hypergeometric functions

2021 ◽  
Vol 33 (1) ◽  
pp. 1-22
Author(s):  
D. Artamonov
Keyword(s):  
Explicit Formula ◽  
Lie Algebra ◽  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Content Type

The Clebsh–Gordan coefficients for the Lie algebra g l 3 \mathfrak {gl}_3 in the Gelfand–Tsetlin base are calculated. In contrast to previous papers, the result is given as an explicit formula. To obtain the result, a realization of a representation in the space of functions on the group G L 3 GL_3 is used. The keystone fact that allows one to carry the calculation of Clebsh–Gordan coefficients is the theorem that says that functions corresponding to the Gelfand–Tsetlin base vectors can be expressed in terms of generalized hypergeometric functions.

scholarly journals Two dimensional Laplace transforms of generalized hypergeometric functions

1988 ◽  
Vol 11 (1) ◽  
pp. 167-175 ◽  
Author(s):  
R. S. Dahiya ◽  
I. H. Jowhar
Keyword(s):  
Hypergeometric Functions ◽  
Laplace Transforms ◽  
Two Dimensional ◽  
Generalized Hypergeometric Functions

The object of this paper is to obtain new operational relations between the original and the image functions that involve generalized hypergeometricG-functions.

New subclasses of analytic functions defined by convolution involving the hypergeometric function and the Owa–Srivastava operator

2019 ◽  
Vol 26 (3) ◽  
pp. 449-458
Author(s):  
Khalida Inayat Noor ◽  
Rashid Murtaza ◽  
Janusz Sokół
Keyword(s):  
Hypergeometric Function ◽  
Convolution Operator ◽  
Analytic Functions ◽  
Hypergeometric Functions ◽  
Integral Operators ◽  
Generalized Hypergeometric Functions ◽  
Convolution Properties ◽  
Appl Math ◽  
Inclusion Results

Abstract In the present paper we introduce a new convolution operator on the class of all normalized analytic functions in {|z|<1} , by using the hypergeometric function and the Owa–Srivastava operator {\Omega^{\alpha}} defined in [S. Owa and H. M. Srivastava, Univalent and starlike generalized hypergeometric functions, Canad. J. Math. 39 1987, 5, 1057–1077]. This operator is a generalization of the operators defined in [S. K. Lee and K. M. Khairnar, A new subclass of analytic functions defined by convolution, Korean J. Math. 19 2011, 4, 351–365] and [K. I. Noor, Integral operators defined by convolution with hypergeometric functions, Appl. Math. Comput. 182 2006, 2, 1872–1881]. Also we introduce some new subclasses of analytic functions using this operator and we discuss some interesting results, such as inclusion results and convolution properties. Our results generalize the results of [S. K. Lee and K. M. Khairnar, A new subclass of analytic functions defined by convolution, Korean J. Math. 19 2011, 4, 351–365].

scholarly journals A New Identity for Generalized Hypergeometric Functions and Applications

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 12 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Wolfram Koepf
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Summation Formulas ◽  
Published Paper

We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38).

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