scholarly journals Some expansions of hypergeometric functions in series of hypergeometric functions

1976 ◽  
Vol 17 (1) ◽  
pp. 17-21 ◽  
Author(s):  
H. M. Srivastava ◽  
Rekha Panda
Keyword(s):  
Analytic Continuation ◽  
Hypergeometric Functions ◽  
Present Note ◽  
Expansion Formula ◽  
In Series ◽  
Image Position

Throughout the present note we abbreviate the set of p parameters a1,…,ap by (ap), with similar interpretations for (bq), etc. Also, by [(ap)]m we mean the product , where [λ]m = Г(λ + m)/ Г(λ), and so on. One of the main results we give here is the expansion formula(1)which is valid, by analytic continuation, when, p,q,r,s,t and u are nonnegative integers such that p+r < q+s+l (or p+r = q+s+l and |zω| <1), p+t < q+u (or p + t = q + u and |z| < 1), and the various parameters including μ are so restricted that each side of equation (1) has a meaning.

Expansions of Generalized Hypergeometric Functions in Series of Products of Generalized Whittaker Functions

1962 ◽  
Vol 58 (2) ◽  
pp. 239-243 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Hypergeometric Function ◽  
Hypergeometric Functions ◽  
Whittaker Function ◽  
Generalized Hypergeometric Function ◽  
Whittaker Functions ◽  
Generalized Hypergeometric Functions ◽  
In Series ◽  
Image Position

In a previous paper (l) in this journal L. J. Slater gave expansions of the generalized Whittaker functions . She gave this name to the generalized hypergeometric function in the sense that it is a generalization of the well-known Whittaker function . In this paper series of products of generalized Whittaker functions will be evaluated in terms of such functions or in terms of generalized hypergeometric functions pFp(x). These expansions are These formulae will be proved in § 2 and particular cases will be given in § 3.

scholarly journals XXV.—The Generating Function of the Reciprocal of a Determinant

1905 ◽  
Vol 40 (3) ◽  
pp. 615-629
Author(s):  
Thomas Muir
Keyword(s):  
Generating Function ◽  
Specific Character ◽  
Partial Fraction ◽  
Homogeneous Functions ◽  
Series Form ◽  
General Proposition ◽  
In Series ◽  
Image Position ◽  
The Given

(1) This is a subject to which very little study has been directed. The first to enunciate any proposition regarding it was Jacobi; but the solitary result which he reached received no attention from mathematicians,—certainly no fruitful attention,—during seventy years following the publication of it.Jacobi was concerned with a problem regarding the partition of a fraction with composite denominator (u1 − t1) (u2 − t2) … into other fractions whose denominators are factors of the original, where u1, u2, … are linear homogeneous functions of one and the same set of variables. The specific character of the partition was only definable by viewing the given fraction (u1−t1)−1 (u2−t2)−1…as expanded in series form, it being required that each partial fraction should be the aggregate of a certain set of terms in this series. Of course the question of the order of the terms in each factor of the original denominator had to be attended to at the outset, since the expansion for (a1x+b1y+c1z−t)−1 is not the same as for (b1y+c1z+a1x−t)−1. Now one general proposition to which Jacobi was led in the course of this investigation was that the coefficient ofx1−1x2−1x3−1…in the expansion ofy1−1u2−1u3−1…, whereis |a1b2c3…|−1, provided that in energy case the first term of uris that containing xr.

scholarly journals On certain infinite integrals involving Struve functions and parabolic cylinder functions

1946 ◽  
Vol 7 (4) ◽  
pp. 171-173 ◽  
Author(s):  
S. C. Mitra
Keyword(s):  
Present Note ◽  
Parabolic Cylinder ◽  
Parabolic Cylinder Functions ◽  
Struve Functions ◽  
Image Position

The object of the present note is to obtain a number of infinite integrals involving Struve functions and parabolic cylinder functions. 1. G. N. Watson(1) has proved thatFrom (1)follows provided that the integral is convergent and term-by-term integration is permissible. A great many interesting particular cases of (2) are easily deducible: the following will be used in this paper.

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.

A result on hypergeometric functions

1967 ◽  
Vol 63 (1) ◽  
pp. 179-182 ◽  
Author(s):  
Samir Kumar Bhattacharya
Keyword(s):  
Hypergeometric Functions ◽  
Image Position

Summary. The main result of the present paper is contained in the following: Theorem ℜ α ≠ −1, −2, −3,…where2F1[.,.;.;.] and 1f1[.;.;.] (l).

Sophocles, ‘Electra’ 1205–10

1973 ◽  
Vol 19 ◽  
pp. 45-46
Author(s):  
R. D. Dawe
Keyword(s):  
Present Note ◽  
Greek Tragedy ◽  
Image Position

The attribution of lines to different speakers in Greek tragedy is a matter on which MSS have notoriously little authority. As for Electra itself, there are at least three places where the name of the heroine has been incorrectly added in some or all MSS. In my Studies in the Text of Sophocles, I, 198, I list these places and suggest that the same error has happened at a fourth place, viz. 1323. The purpose of the present note is to suggest that at El. 1205–10 the same mistake has happened yet again.The situation is that Electra is holding the urn which she falsely believes to contain the ashes of her dead brother, Orestes. But Orestes is alive, and before her at this very moment. He is trying to persuade her to give up the urn. If the text before us had been preserved in a MS devoid of ascriptions to speakers, no one would have been so perverse as to do what all MSS and editors do in fact do, namely attribute the words οὔ φημ᾿ ἐάσειν to Orestes.

scholarly journals On general curves lying on a quadric

1927 ◽  
Vol 1 (1) ◽  
pp. 19-30 ◽  
Author(s):  
H. F. Baker
Keyword(s):  
Present Note ◽  
Rational Curves ◽  
Stationary Points ◽  
Image Position

Introduction. The present note, though in continuation of the preceding one dealing with rational curves, is written so as to be independent of this. It is concerned to prove that if a curve of order n, and genus p, with k cusps, or stationary points, lying on a quadric, Ω, in space of any number of dimensions, is such that itself, its tangents, its osculating planes, … , and finally its osculating (h – 1)-folds, all lie on the quadric Ω, then the number of its osculating h-folds which lie on the quadric isTwo proofs of this result are given, in §§ 4 and 5.

scholarly journals On the application of the operational calculus to the expansion of a function in a series of Legendre's functions of the second kind

1942 ◽  
Vol 7 (1) ◽  
pp. 1-2
Author(s):  
D. P. Banerjee
Keyword(s):  
Analytic Function ◽  
Present Note ◽  
Operational Calculus ◽  
Integral Function ◽  
Legendre Functions ◽  
Image Position

In the present note we shall obtain the expansion in a series of Legendre functions of the second kind of an integral function φ (ω) represented by Laplace's integralwhere f (x) is an analytic function of x, regular in the circlewhere an are constants and qn (ω) = in+1Qn (iω).

A System of Linear Equations with an Infinity of Unknowns

1924 ◽  
Vol 22 (3) ◽  
pp. 282-286
Author(s):  
E. C. Titchmarsh
Keyword(s):  
Present Note ◽  
Linear Equations ◽  
The Other ◽  
System Of Linear Equations ◽  
Other Hand ◽  
Image Position ◽  
Monotonic Character

I have collected in the present note some theorems regarding the solution of a certain system of linear equations with an infinity of unknowns. The general form of the equations isthe numbers a1, a2, … c1, c2, … being given. Equations of this type are of course well known; but in studying them it is generally assumed that the series depend for convergence on the convergence-exponent of the sequences involved, e.g. that and are convergent. No assumptions of this kind are made here, and in fact the series need not be absolutely convergent. On the other hand rather special assumptions are made with regard to the monotonic character of the sequences an and cn.

scholarly journals Polynomially bounded multisequences and analytic continuation

1982 ◽  
Vol 23 (1) ◽  
pp. 41-52
Author(s):  
Daniel J. Troy
Keyword(s):  
Power Series ◽  
Analytic Continuation ◽  
Linear Functional ◽  
Necessary And Sufficient Condition ◽  
Convergence Criteria ◽  
Dimensional Torus ◽  
Unit Factor ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Class Of Functions

Given a polynomially bounded multisequence {fm}, where m = (m1, …, mk) ∈ ℤk, we will consider 2k power series in exp(iz1), …, exp(izk), each representing a holomorphic function within its domain of convergence. We will consider this same multisequence as a linear functional on a class of functions defined on the k-dimensional torus by a Fourier series, , with the proper convergence criteria. We shall discuss the relationships that exist between the linear functional properties of the multisequence and the analytic continuation of the holomorphic functions. With this approach we show that a necessary and sufficient condition that the multisequence be given by a polynomial is that each of the power series represents, up to a unit factor, the same function that is entire in the variables

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