A094368 Triangle M(k,n) read by rows: coefficients of Meixner polynomials.
1, 1, -1, 1, -5, 1, -14, 9, 1, -30, 89, 1, -55, 439, -225, 1, -91, 1519, -3429, 1, -140, 4214, -24940, 11025, 1, -204, 10038, -122156, 230481, 1, -285, 21378, -463490, 2250621, -893025, 1, -385, 41778, -1467290, 14466221, -23941125, 1, -506
Offset: 1
Examples
z, z^2 - 1, z^3 - 5*z, z^4 - 14*z^2 + 9, z^5 - 30*z^3 + 89*z, z^6 - 55*z^4 + 439*z^2 - 225, z^7 - 91*z^5 + 1519*z^3 - 3429*z, z^8 - 140*z^6 + 4214*z^4 - 24940*z^2 + 11025, z^9 - 204*z^7 + 10038*z^5 - 122156*z^3 + 230481*z,
Links
- Paul L. Butzer and Tom H. Koornwinder, Josef Meixner: His life and his orthogonal polynomials, Indagationes Mathematicae, Volume 30, Issue 1, January 2019, Pages 250-264.
- Dominique Foata, Combinatoire des identités sur les polynomes de Meixner, Sem. Loth. de Comb. B06c (1982).
- Nicolas Loizeau, Berislav Buča, and Dries Sels, Opening Krylov space to access all-time dynamics via dynamical symmetries, arXiv:2503.07403 [quant-ph], 2025. See p. 9.
- Josef Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. Lond. Math. Soc. 9 (1934), 6-13.
- M. Micu, Continuous Hahn polynomials, J. Math. Phys. 34 (3) (1993), 1197-1205.
- Eric Weisstein's World of Mathematics, Meixner Polynomial of the Second Kind
Formula
Recurrence: M(0, z) = 1, M(1, z) = z, M(n+1, z) = z*M(n, z) - n^2*M(n-1, z).
G.f.: exp(z*arctan(x)) / sqrt(1+x^2).
The n-th (unsigned) row polynomial R(n, x) = (-i)^n * M(n, i*x) = n!*Sum_{k = 0..n} 2^k*binomial(n, k)*binomial(x/2 - 1/2, k). - Peter Bala, Mar 10 2024