Jet Wimp

Expansions of hypergeometric functions in hypergeometric functions

In (1) Luke gave an expansion of the confluent hypergeometric func- tion in terms of the modified Bessel functions I,(z). The existence of other, similar expansions implied that more general expansions might exist. Such was the case. Here multiplication type expansions of low-order hypergeometric functions in terms of other hypergeometric functions are generalized by Laplace transform techniques. 1. General Expansions. The generalized hypergeometric function pFq(z), (2), is defined by ~~~~~~~~~~~~P KF~~~~~~~~~(Z) = Z) ~~~~~~~~~~~~~~~~~~~~~~~~~~~ I~~~~~I (ai)~

Computing the Hilbert transform of a Jacobi weight function

We discuss the evaluation of the Hilbert transformf−11(t-ξ)−1w(α, β)(t)dt,−1<ξ<1, of the Jacobi weight functionw(α, β)(t)=(1−t)α)(1+t)β by analytic and numerical means and also comment on the recursive computation of the quantitiesf−11)(t−ξ)−1πn(t;w(α, β))w(α, β)(t)dt,n=0, 1, 2, ..., whereπn(·;w(α, β)) is the Jacobi polynomial of degreen.

Some transformations of monotone sequences

In this paper, we discuss a class of methods for summing sequences which are generalizations of a method due to Salzer. The methods are not regular, and in contrast to the classical regular methods, seem to work best on sequences which are monotone. In our main theorem, we determine a class of convergent sequences for which the methods yield sequences which converge to the same sum.

Polynomial expansions of Bessel functions and some associated functions

Abstract : IN THIS REPORT WE FIRST DETERMINE REPRESENTATIONS FOR THE Anger-Weber functions (ax) and (ax) in series of symmetric Jacobi polynomials. (These include Legendre and Chebyshev polynomials as special cases.) If is an integer, these become expansions for the Bessel function of the first kind, since n(ax) = Jn(ax). Next, corresponding representations are found for (ax) - J (ax). Convenient error bounds are obtained for the Chebyshev cases of the above expansions. In the final section of the report we determine the similar type expansions for the Bessel functions Yn(ax) and Kn(ax).

The asymptotic representation of a class of

and for a, ß, a i, b¡ suitably restricted. (Our analysis will reveal that many of these restrictions may be dropped.) Since fi\x) has an asymptotic representation in descending powers of \x, (2) may be interpreted as a summation process which converts the generally divergent expansion into a convergent one. Important special cases of (2) yield expansions for the confluent hypergeometric function \p(a, c; \x) and Lommel functions. We will treat only the case Q — P — 1 > 0 since the case P + 1 ^ Q may be handled by an elementary analysis. In the former, <E>n(M), as we shall see, has the unusual behavior of exponential decay as n —-> °o f in contrast to the latter case, where

G

>AM) behaves as inverse powers of w!, or at worst (P + 1 = Q), algebraically in n. In Section II, we first prove three lemmas; the first establishes an integral representation for

-functions for large parameter

„(ö)(x); the second estimates for large n a closely related integral, and the third gives the desired asymptotic formula for

Differential-difference properties of hypergeometric polynomials

n(e)(X). Our main theorem follows when we find we can express <Ên(A/)(A) as a linear combination of the functions

Sequence transformations and their applications

„(e)[X exp (wi(Q M 2k))]. Section III is devoted to examples. There are quantities and assumptions about them which occur frequently in this paper, and they will always be as below :

Computation with Recurrence Relations.

We develop differential-difference properties of a class of hypergeometric polynomials which are a generalization of the Jacobi polynomials. The formulas are analogous to known formulas for the classical orthogonal polynomials.