Expansion of the Confluent Hypergeometric Function in Series of Bessel Functions

1959 ◽  
Vol 13 (68) ◽  
pp. 261 ◽  
Author(s):  
Yudell L. Luke
Keyword(s):  
Hypergeometric Function ◽  
Bessel Functions ◽  
Confluent Hypergeometric Function ◽  
In Series

A note on the real zeros of the basic confluent hypergeometric function

1968 ◽  
Vol 64 (3) ◽  
pp. 683-686
Author(s):  
Ramadhar Mishra
Keyword(s):  
Hypergeometric Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Numerical Evaluation ◽  
Confluent Hypergeometric Function ◽  
The Real ◽  
Real Zeros ◽  
In Series

Some years back, Slater (4) discussed the approximations, based on the expansion in series, for the cases 1F1(a; b; x) = 0, when either of b and x or a and x are fixed. These approximations were based essentially on the well-known Newton's method of approximation and were helpful in the numerical evaluation of the small real zeros of the confluent hypergeometric function 1F1(a; b; x;). In this note, we deal with the corresponding problem for the basic confluent hypergeometric function 1Φ1(a; b; x;).

Expansions and asymptotics associated with basic hypergeometric functions and modified q-Bessel functions

2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ahmed Salem
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Confluent Hypergeometric Function ◽  
Basic Hypergeometric Functions

Abstract.In this paper, the expansion of a basic confluent hypergeometric function in terms of an incomplete

scholarly journals MORE SPECIAL FUNCTIONS TRAPPED

2013 ◽  
Vol 24 (02) ◽  
pp. 1350004
Author(s):  
CHARLES SCHWARTZ
Keyword(s):  
Hypergeometric Function ◽  
Bessel Functions ◽  
Special Functions ◽  
Mathematical Physics ◽  
Confluent Hypergeometric Function ◽  
Trapezoidal Rule ◽  
Integral Representations ◽  
Gamma Functions ◽  
Incomplete Gamma Functions

We extend the technique of using the trapezoidal rule for efficient evaluation of the special functions of mathematical physics given by integral representations. This technique was recently used for Bessel functions, and here we treat incomplete gamma functions and the general confluent hypergeometric function.

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.

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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