Expansions of generalized Whittaker functions

1954 ◽  
Vol 50 (4) ◽  
pp. 628-631 ◽  
Author(s):  
L. J. Slater
Keyword(s):  
Bessel Functions ◽  
Direct Proof ◽  
Whittaker Function ◽  
Whittaker Functions

ABSTRACTIn this paper I give a direct proof of a new expansion of the Whittaker function M(a; b; x) as a series of Bessel functions, and some generalizations of this result for the functions 2F2(x) and 4F4(x).

Expansions of certain Whittaker functions

1960 ◽  
Vol 56 (3) ◽  
pp. 233-239 ◽  
Author(s):  
A. S. Meligy
Keyword(s):  
Bessel Functions ◽  
Whittaker Function ◽  
Wave Functions ◽  
Whittaker Functions ◽  
Angular Momenta

ABSTRACTAn expansion of the Whittaker function Wk, m(z) in terms of Bessel functions is given for 2m an integer or zero. Simple formulae are derived for the cases of m = 0, ½, 1 and . The case m = 0 provides expansions for the exponential, sine, cosine and similar integrals. The cases m = ½ and provide expansions for the irregular Coulomb wave functions having angular momenta zero and one.

The Inverse Laplace Transform of the Product of Two Whittaker Functions

1962 ◽  
Vol 58 (4) ◽  
pp. 580-582 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Laplace Transform ◽  
Whittaker Function ◽  
Inverse Laplace Transform ◽  
Whittaker Functions ◽  
Original Function ◽  
The Laplace Transform ◽  
Image Position

The object of this paper is to obtain the original function of which the Laplace transform (l) is the productwhere, as usual, p is complex, n is any positive integer, and Wk, m(z) is the Whittaker function defined by the equationIn § 2 it will be shown that this original function iswhere the symbol Δ(n; α) represents the set of parameters

scholarly journals WHITTAKER FUNCTIONS AND DEMAZURE CHARACTERS

2017 ◽  
Vol 18 (4) ◽  
pp. 759-781
Author(s):  
Kyu-Hwan Lee ◽  
Cristian Lenart ◽  
Dongwen Liu
Keyword(s):  
Explicit Formula ◽  
Transition Matrix ◽  
Whittaker Function ◽  
Whittaker Functions ◽  
Induced Representation ◽  
Demazure Characters

In this paper, we consider how to express an Iwahori–Whittaker function through Demazure characters. Under some interesting combinatorial conditions, we obtain an explicit formula and thereby a generalization of the Casselman–Shalika formula. Under the same conditions, we compute the transition matrix between two natural bases for the space of Iwahori fixed vectors of an induced representation of a $p$-adic group; this corrects a result of Bump–Nakasuji.

Derivation of Green-type, transitional, and uniform asymptotic expansions from differential equations II. Whittaker functions W k,m for large k , and for large | K 2 – m 2 |

1964 ◽  
Vol 281 (1384) ◽  
pp. 111-129 ◽  
Keyword(s):  
Differential Equations ◽  
Asymptotic Expansions ◽  
Mellin Transform ◽  
Bessel Functions ◽  
Parabolic Cylinder ◽  
Whittaker Functions ◽  
Transform Method ◽  
Special Cases ◽  
Uniform Expansions ◽  
Uniform Asymptotic Expansions

The method for deriving Green-type asymptotic expansions from differential equations, introduced in I and illustrated therein by detailed calculations on modified Bessel functions, is applied to Whittaker functions W k,m , first for large k , and then for large |k 2 —m 2 |. Following the general theory of I, combination of this procedure with the Mellin transform method yields asymptotic expansions valid in transitional regions, and general uniform expansions. Weber parabolic cylinder and Poiseuille functions are examined as important special cases.

Crystals and p-adic Integration

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Partition Function ◽  
Generating Function ◽  
Whittaker Function ◽  
Character Formula ◽  
Whittaker Functions ◽  
Unipotent Subgroup ◽  
Enveloping Algebra ◽  
Quantized Enveloping Algebra ◽  
Kostant Partition Function ◽  
Weyl Character Formula

This chapter describes the properties of Kashiwara's crystal and its role in unipotent p-adic integrations related to Whittaker functions. In many cases, integrations of representation theoretic import over the maximal unipotent subgroup of a p-adic group can be replaced by a sum over Kashiwara's crystal. Partly motivated by the crystal description presented in Chapter 2 of this book, this perspective was advocated by Bump and Nakasuji. Later work by McNamara and Kim and Lee extended this philosophy yet further. Indeed, McNamara shows that the computation of the metaplectic Whittaker function is initially given as a sum over Kashiwara's crystal. The chapter considers Kostant's generating function, the character of the quantized enveloping algebra, and its association with Kashiwara's crystal, along with the Kostant partition function and the Weyl character formula.

scholarly journals Laplace transform of certain functions with applications

2000 ◽  
Vol 23 (2) ◽  
pp. 99-102
Author(s):  
M. Aslam Chaudhry
Keyword(s):  
Laplace Transform ◽  
Bessel Functions ◽  
Whittaker Functions ◽  
Special Cases ◽  
The Laplace Transform ◽  
Special Case

The Laplace transform of the functionstν(1+t)β,Reν>−1, is expressed in terms of Whittaker functions. This expression is exploited to evaluate infinite integrals involving products of Bessel functions, powers, exponentials, and Whittaker functions. Some special cases of the result are discussed. It is also demonstrated that the famous identity∫0∞sin (ax)/x dx=π/2is a special case of our main result.

scholarly journals NEW ANALYSIS OF WITTAKER FUNCTIONS

1990 ◽  
Vol 21 (4) ◽  
pp. 303-326
Author(s):  
S. F. RAGAB
Keyword(s):  
Bessel Functions ◽  
Whittaker Functions

Integrals involving products of two Whittaker functions and Bessel functions are evaluated in §§ 3,4. Also the integrals \[ \int_0^\infty t^{\rho-1}W_{k_1m}(t)W_{-k_1m}(t)W_{\mu, \nu}\left(\frac{2iz}{t}\right)W_{\mu, \nu}\left(-\frac{2iz}{t}\right)\ dt\] and \[ \int_0^\infty t^{\rho-1}W_{k_1m}(t)W_{-k_1m}(t)W_{\mu, \nu}(2zt)W_{-\mu, \nu}(2zt)\ dt\] are evaluated in § 5 while in § 6 integrals involving the product of three Whittaker functions are established.

UNIFORM ASYMPTOTIC APPROXIMATIONS FOR THE WHITTAKER FUNCTIONS Mκ,iμ(z) AND Wκ,iμ(z)

2003 ◽  
Vol 01 (02) ◽  
pp. 199-212 ◽  
Author(s):  
T. M. DUNSTER
Keyword(s):  
Hypergeometric Functions ◽  
Turning Point ◽  
Bessel Functions ◽  
Asymptotic Approximations ◽  
Whittaker Functions ◽  
Modified Bessel Functions ◽  
Airy Functions ◽  
Double Pole ◽  
Confluent Hypergeometric Functions ◽  
Uniform Reduction

Uniform asymptotic approximations are obtained for the Whittaker's confluent hypergeometric functions Mκ, iμ(z) and Wκ, iμ(z), where κ, μ and z are real. Three cases are considered, and when taken together, result in approximations which are valid for κ → ∞ uniformly for 0 ≤ μ < ∞, 0 < z < ∞, and also for μ → ∞ uniformly for 0 ≤ κ < ∞, 0 < z < ∞. The results are obtained by an application of general asymptotic theories for differential equations either having a coalescing turning point and double pole with complex exponent, or a fixed simple turning point. The resulting approximations achieve a uniform reduction of free variables from three to two, and involve either modified Bessel functions or Airy functions. Explicit error bounds are available for all the approximations.

Whittaker Functions

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Dirichlet Series ◽  
Weyl Group ◽  
Eisenstein Series ◽  
Local Field ◽  
Whittaker Function ◽  
Global Field ◽  
Whittaker Functions ◽  
Schur Polynomials ◽  
Multiple Dirichlet Series ◽  
Adele Group

This chapter shows that Weyl group multiple Dirichlet series are expected to be Whittaker coefficients of metaplectic Eisenstein series. The fact that Whittaker coefficients of Eisenstein series reduce to the crystal description that was given in Chapter 2 is proved for Type A. On the adele group, the corresponding local computation reduces to the evaluation of a type of λ‎-adic integral. These were considered by McNamara, who reduced the integrals to sums over crystals by a very interesting method. A full treatment of this topic is outside the scope of this work, but it is introduced in this chapter by considering the case where n = 1. In this chapter, F is used to denote a nonarchimedean local field and F to denote a global field. The values of the Whittaker function are Schur polynomials multiplied by the normalization constant.

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