scholarly journals EXTENSION OF EXTENDED BETA, HYPERGEOMETRIC AND CONFLUENT HYPERGEOMETRIC FUNCTIONS

2014 ◽  
Vol 36 (2) ◽  
pp. 357-385 ◽  
Author(s):  
Junesang Choi ◽  
Arjun K. Rathie ◽  
Rakesh K. Parmar
Keyword(s):  
Hypergeometric Functions ◽  
Confluent Hypergeometric Functions

scholarly journals On the birthday problem: some generalizations and applications

2003 ◽  
Vol 2003 (60) ◽  
pp. 3827-3840 ◽  
Author(s):  
P. N. Rathie ◽  
P. Zörnig
Keyword(s):  
Probability Distribution ◽  
Random Graph ◽  
Generating Functions ◽  
Hypergeometric Functions ◽  
Exact Probability ◽  
Confluent Hypergeometric Functions ◽  
Birthday Problem ◽  
Exact Probability Distribution

We study the birthday problem and some possible extensions. We discuss the unimodality of the corresponding exact probability distribution and express the moments and generating functions by means of confluent hypergeometric functionsU(−;−;−)which are computable using the software Mathematica. The distribution is generalized in two possible directions, one of them consists in considering a random graph with a single attracting center. Possible applications are also indicated.

scholarly journals FOUR-PARTICLE ANYON EXCITON: BOSON APPROXIMATION

1995 ◽  
Vol 09 (02) ◽  
pp. 123-133 ◽  
Author(s):  
M. E. Portnoi ◽  
E. I. Rashba
Keyword(s):  
Angular Momentum ◽  
Electron Density ◽  
Ground State ◽  
Hypergeometric Functions ◽  
Energy Levels ◽  
Confluent Hypergeometric Functions ◽  
Full Symmetry ◽  
Hole Confinement ◽  
Boson Approximation

A theory of anyon excitons consisting of a valence hole and three quasielectrons with electric charges –e/3 is presented. A full symmetry classification of the k = 0 states is given, where k is the exciton momentum. The energy levels of these states are expressed by quadratures of confluent hypergeometric functions. It is shown that the angular momentum L of the exciton ground state depends on the distance between the electron and hole confinement planes and takes the values L = 3n, where n is an integer. With increasing k the electron density shows a spectacular splitting on bundles. At first a single anyon splits off of the two-anyon core, and finally all anyons become separated.

scholarly journals IX.—The Confluent Hypergeometric Functions of Two Variables

1922 ◽  
Vol 41 ◽  
pp. 73-96 ◽  
Author(s):  
Pierre Humbert
Keyword(s):  
Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Functions Of Two Variables

This memoir is devoted to the study of certain new functions, which may be regarded as limiting cases of the “hypergeometric functions of two variables” discovered by Appell.

scholarly journals ON SOME NEW Pδ-TRANSFORMS OF KUMMER'S CONFLUENT HYPERGEOMETRIC FUNCTIONS

2019 ◽  
pp. 373
Author(s):  
Rakesh K. Parmar ◽  
Vivek Rohira ◽  
Arjun K. Rathie
Keyword(s):  
Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Special Cases ◽  
Summation Theorem

The aim of our paper is to present Pδ -transforms of the Kummer’s confluent hypergeometric functions by employing the generalized Gauss’s second summation the-orem, Bailey’s summation theorem and Kummer’s summation theorem obtained earlier by Lavoie, Grondin and Rathie [9]. Relevant connections of certain special cases of the main results presented here are also pointed out.

scholarly journals Expressions of the Laguerre polynomial and some other special functions in terms of the generalized Meijer $ G $-functions

2021 ◽  
Vol 6 (11) ◽  
pp. 11631-11641
Author(s):  
Syed Ali Haider Shah ◽  
◽  
Shahid Mubeen
Keyword(s):  
Hypergeometric Functions ◽  
Laguerre Polynomial ◽  
Special Functions ◽  
Laguerre Polynomials ◽  
Exponential Functions ◽  
Confluent Hypergeometric Functions ◽  
Logarithmic Functions

<abstract><p>In this paper, we investigate the relation of generalized Meijer $ G $-functions with some other special functions. We prove the generalized form of Laguerre polynomials, product of Laguerre polynomials with exponential functions, logarithmic functions in terms of generalized Meijer $ G $-functions. The generalized confluent hypergeometric functions and generalized tricomi confluent hypergeometric functions are also expressed in terms of the generalized Meijer $ G $-functions.</p></abstract>

scholarly journals Uniform asymptotic expansions for the Whittaker functions M κ , μ ( z ) and W κ , μ ( z ) with μ large

2021 ◽  
Vol 477 (2252) ◽  
pp. 20210360
Author(s):  
T. M. Dunster
Keyword(s):  
Analytic Continuation ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Hypergeometric Functions ◽  
Whittaker Functions ◽  
Airy Functions ◽  
Elementary Functions ◽  
Confluent Hypergeometric Functions ◽  
Uniform Asymptotic Expansions

Uniform asymptotic expansions are derived for Whittaker’s confluent hypergeometric functions M κ , μ ( z ) and W κ , μ ( z ) , as well as the numerically satisfactory companion function W − κ , μ ( z   e − π i ) . The expansions are uniformly valid for μ → ∞ , 0 ≤ κ / μ ≤ 1 − δ < 1 and 0 ≤ arg ⁡ ( z ) ≤ π . By using appropriate connection and analytic continuation formulae, these expansions can be extended to all unbounded non-zero complex z . The approximations come from recent asymptotic expansions involving elementary functions and Airy functions, and explicit error bounds are either provided or available.

The magneto-hydrodynamic boundary layer in the two-dimensional steady flow past a semi-infinite flat plate II. The Falkner-Skan problem

1963 ◽  
Vol 273 (1355) ◽  
pp. 509-517 ◽  
Keyword(s):  
Flat Plate ◽  
Iterative Procedure ◽  
Hypergeometric Functions ◽  
Close Agreement ◽  
Adverse Pressure Gradient ◽  
Iterative Solution ◽  
Two Dimensional ◽  
Confluent Hypergeometric Functions ◽  
First Case ◽  
Hydrodynamic Boundary Layer

The problem investigated is the flow of a viscous liquid past a semi-infinite flat plate against an adverse pressure gradient. The method used is an iterative method suggested in a paper by Weyl. To start the iterative procedure a function is chosen which satisfies some of the boundary conditions and by using this function the first iterative solution has been obtained analytically in terms of confluent hypergeometric functions. Two different starting functions have been considered. In the first case it has been found possible to compare the results obtained with the well-known Hartree numerical solution and even at the first iteration close agreement is achieved. In the second case, the first iterative solution behaves correctly at infinity but the agreement with Hartree ’s solution is not as good as it is in the first case.

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