In this article we study in detail the expansion of the Riemann $\Xi$ function in Meixner--Pollaczek polynomials. We obtain explicit formulas, recurrence relation and asymptotic expansion for the coefficients and investigate the zeros of the partial sums.

MSC Classifications:

41A10, 11M26, 33C45 show english descriptions Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10}
Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions] 41A10 - Approximation by polynomials {For approximation by trigonometric polynomials, see 42A10}
11M26 - Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
33C45 - Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) [See also 42C05 for general orthogonal polynomials and functions]

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