Uniform asymptotic expansions for the discrete Chebyshev polynomials (Q2905695)

Asymptotic expansions for the discrete Chebyshev polynomials, \(t_n(x,N)= n! \Delta^n \binom{x}{n} \binom{x-N}{n}\), as \(x \rightarrow \infty\), are derived. Specifically, letting \(a=x/N\) and \(b=n/N\), the authors obtain the expansions NEWLINE\[NEWLINE t_n(x, N+1) \sim \frac{(-1)^n\Gamma(n+N+2)N^{-aN}e^{N\gamma}}{\Gamma(n+1)\Gamma(N-n+1)\Gamma(-aN+1)}\sum_{l=0}^\infty \frac{c_l}{N^{l+1/2}} NEWLINE\]NEWLINE for \(a<0\), where \(\gamma\) is a constant and NEWLINE\[NEWLINE c_l=\sum_{m=0}^l a_{l-m, 2m}\Gamma\left(\frac{2m+1}{2}\right), NEWLINE\]NEWLINE and the coefficients \(a_{n,m}\) are the coefficients of Maclaurin expansions of functions that appear in an integral representation of the discrete Chebyshev polynomials. For \(0 \leq a \leq 1/2\), the expansion takes the form NEWLINE\[NEWLINE\begin{multlined} t_n(x, N+1) \sim \frac{(-1)^n \Gamma(n+N+2)e^{N\gamma}}{\Gamma(n+1)\Gamma(N-n+1)} \bigg[ \mathbf{M}(an+1, 1, \eta N) \sum_{l=0}^\infty \frac{c_l}{N^{l+1/2}} \\ + \mathbf{M'}(aN+1, 1, \eta N) \sum_{l=0}^\infty \frac{d_l}{N^{l+1/2}} \bigg], \end{multlined}NEWLINE\]NEWLINE where \(\gamma\) and \(\eta\) are constants, \(c_l\) is as above, \(d_l\) is very similar to \(c_l\), and \(\mathbf{M}\) is a confluent hypergeometric function defined by NEWLINE\[NEWLINE \mathbf{M}(d,c,z)=\frac{\Gamma(1+d-c)}{2\pi i\Gamma(d)}\int_{\gamma_1} u^{d-1} (u-1)^{c-d-1}e^{zu}\, du, NEWLINE\]NEWLINE where \(\gamma_1\) is a specific closed curve in the complex plane. The derivative \(\mathbf{M}'(d,c,z)\) is taken with respect to the second variable.NEWLINENEWLINETo prove the result, the authors obtain double integral representations for the polynomials, and then use the method of steepest descent to obtain canonical integral representations for the polynomials. A detailed examination of the analyticity of the mappings and functions used to obtain these canonical integral representations is included. An asymptotic expansion is obtained using integration by parts and Watson's lemma. Error estimates are proven, and additional results of expanding the region of validity for the asymptotic expansion, and asymptotic approximation for the largest and smallest zeros of \(t_n(x,N+1)\) are also proven.