scholarly journals The Bateman Functions Revisited after 90 Years—A Survey of Old and New Results

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1273
Author(s):  
Alexander Apelblat ◽  
Armando Consiglio ◽  
Francesco Mainardi
Keyword(s):  
Hypergeometric Function ◽  
Laplace Transforms ◽  
Confluent Hypergeometric Function ◽  
Integral Function ◽  
Basic Properties ◽  
Differential Relations

The Bateman functions and the allied Havelock functions were introduced as solutions of some problems in hydrodynamics about ninety years ago, but after a period of one or two decades they were practically neglected. In handbooks, the Bateman function is only mentioned as a particular case of the confluent hypergeometric function. In order to revive our knowledge on these functions, their basic properties (recurrence functional and differential relations, series, integrals and the Laplace transforms) are presented. Some new results are also included. Special attention is directed to the Bateman and Havelock functions with integer orders, to generalizations of these functions and to the Bateman-integral function known in the literature.

scholarly journals Generalized Laplace transform with matrix variables

1987 ◽  
Vol 10 (3) ◽  
pp. 503-511
Author(s):  
R. M. Joshi ◽  
J. M. C. Joshi
Keyword(s):  
Laplace Transform ◽  
Hypergeometric Function ◽  
Hankel Transform ◽  
Laplace Transforms ◽  
Confluent Hypergeometric Function ◽  
Symmetric Matrices ◽  
The Real ◽  
Matrix Argument ◽  
Alpha Beta

In the present paper we have extended generalized Laplace transforms of Joshi to the space ofm×msymmetric matrices using the confluent hypergeometric function of matrix argument defined by Herz as kernel. Our extension is given byg(z)=Γm(α)Γm(β)∫∧>01F1(α:β:−∧z) f(∧)d∧The convergence of this integral under various conditions has also been discussed. The real and complex inversion theorems for the transform have been proved and it has also been established that Hankel transform of functions of matrix argument are limiting cases of the generalized Laplace transforms.

scholarly journals A study of generalized summation theorems for the series 2F1 with an applications to Laplace transforms of convolution type integrals involving Kummer's functions 1F1

2018 ◽  
Vol 12 (1) ◽  
pp. 257-272
Author(s):  
Gradimir Milovanovic ◽  
Rakesh Parmar ◽  
Arjun Rathie
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Integral Transform ◽  
Laplace Transforms ◽  
Confluent Hypergeometric Function ◽  
Convolution Type ◽  
Product Theorem ◽  
Confluent Hypergeometric Functions

Motivated by recent generalizations of classical theorems for the series 2F1 [Integral Transform. Spec. Funct. 229(11), (2011), 823-840] and interesting Laplace transforms of Kummer's confluent hypergeometric functions obtained by Kim et al. [Math. Comput. Modelling 55 (2012), 1068-1071], first we express generalized summations theorems in explicit forms and then by employing these, we derive various new and useful Laplace transforms of convolution type integrals by using product theorem of the Laplace transforms for a pair of Kummer's confluent hypergeometric function.

scholarly journals On p-Cauchy numbers

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2731-2742 ◽  
Author(s):  
Mourad Rahmani
Keyword(s):  
Hypergeometric Function ◽  
Recurrence Relation ◽  
Confluent Hypergeometric Function ◽  
Bernoulli Numbers ◽  
Basic Properties ◽  
New Family ◽  
Cauchy Numbers ◽  
Three Term Recurrence Relation

In this paper we define a new family of p-Cauchy numbers by means of the confluent hypergeometric function. We establish some basic properties. As consequence, a number of algorithms based on three-term recurrence relation for computing Cauchy numbers of both kinds, Bernoulli numbers of the second kind are derived.

Asymptotics of Sturm-Liouville eigenvalues for problems on a finite interval with one limit-circle singularity, I

1984 ◽  
Vol 99 (1-2) ◽  
pp. 51-70 ◽  
Author(s):  
F. V. Atkinson ◽  
C. T. Fulton
Keyword(s):  
Asymptotic Expansion ◽  
Eigenvalue Problem ◽  
Hypergeometric Function ◽  
Finite Interval ◽  
Confluent Hypergeometric Function ◽  
Limit Circle ◽  
Asymptotic Formulae ◽  
Sturm Liouville

SynopsisAsymptotic formulae for the positive eigenvalues of a limit-circle eigenvalue problem for –y” + qy = λy on the finite interval (0, b] are obtained for potentials q which are limit circle and non-oscillatory at x = 0, under the assumption xq(x)∈L1(0,6). Potentials of the form q(x) = C/xk, 0<fc<2, are included. In the case where k = 1, an independent check based on the limit-circle theory of Fulton and an asymptotic expansion of the confluent hypergeometric function, M(a, b; z), verifies the main result.

A Generalised Kummer-Type Transformation for the pFp(x) Hypergeometric Function

2012 ◽  
Vol 55 (3) ◽  
pp. 571-578
Author(s):  
A. R. Miller ◽  
R. B. Paris
Keyword(s):  
Hypergeometric Function ◽  
Confluent Hypergeometric Function ◽  
Reduction Formula ◽  
Type Transformation

AbstractIn a recent paper, Miller derived a Kummer-type transformation for the generalised hypergeometric function pFp(x) when pairs of parameters differ by unity, by means of a reduction formula for a certain Kampé de Fériet function. An alternative and simpler derivation of this transformation is obtained here by application of the well-known Kummer transformation for the confluent hypergeometric function corresponding to p = 1.

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