The theory of the incomplete gamma functions, as part of the theory of conuent hypergeometric functions, has received its rst systematic exposition by Tricomi in the early 1950s. His own contributions, as well as further advances made thereafter, are surveyed here with particular emphasis on asymptotic expansions, zeros, inequalities, computational methods, and applications.

We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets. Computational schemes are presented to obtain the numerical values of the zeros within high precision. 1991 Mathematics Subject Classification: 33B20, 41A60, 65D20. Keywords & Phrases: Daubechies wavelets, zeros of high degree polynomials, incomplete beta functions, uniform asymptotic expansion. 1. Daubechies wavelets The polynomial formed by the first N terms of the binomial expansion (1 \Gamma y) \GammaN = 1 X k=0 ` k +N \Gamma 1 k ' y k = 1 X k=0 (N) k k! y k that is, the polynomial PN (y) = N \Gamma1 X k=0 ` k +N \Gamma 1 k ' y k = 1 +Ny + N(N + 1) 2 y 2 + : : : + ` 2N \Gamma 2 N \Gamma 1 ' y N \Gamma1 ; (1:1) where (a) k = \Gamma(a+k)=\Gamma(a), plays an important role in the construction of the compactly supp...

We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets. Computational schemes are presented to obtain the numerical values of the zeros within high precision. 1991 Mathematics Subject Classification: 33B20, 41A60, 65D20. Keywords & Phrases: Daubechies wavelets, zeros of high degree polynomials, incomplete beta functions, uniform asymptotic expansion. 1. Daubechies wavelets The polynomial formed by the first N terms of the binomial expansion (1 \Gamma y) \GammaN = 1 X k=0 ` k +N \Gamma 1 k ' y k = 1 X k=0 (N) k k! y k that is, the polynomial PN (y) = N \Gamma1 X k=0 ` k +N \Gamma 1 k ' y k = 1 +Ny + N(N + 1) 2 y 2 + : : : + ` 2N \Gamma 2 N \Gamma 1 ' y N \Gamma1 ; (1:1) where (a) k = \Gamma(a+k)=\Gamma(a), plays an important role in the construction of the compactly supported Daubechies wavelets. There is a close connection between the zeros of PN (y) and the 2N filter coefficients h(n) of the Daubechies wavelet D 2N . For a complete account of the theory we refer to Chapter 6 of Daubechies (1992). In this section we give the details that are relevant to this paper. In later sections we describe the asymptotic methods for obtaining the zeros of PN (y) and discuss methods how to obtain the coefficients h(n) numerically. This paper is an extension of work done by Shen & Strang (1995). 2 1.1. Some properties of Daubechies wavelets We recall that the filter coefficients h(n) define the dilation equation OE(x) = p 2 2N \Gamma1 X n=0 h(n)OE(2x \Gamma n); (1:2) the solution of which is called the scaling function. We take the following normalization of ...