A344909 T(n, k) = numerator([x^k] [z^n] ((1 - i*z)/(1 + i*z))^(i*x)*(1 + z^2)^(-3/4)). Numerators of the coefficients of the symmetric Meixner-Pollaczek polynomials P^(3/4)_{n}(x, Pi/2). Triangle read by rows, T(n, k) for 0 <= k <= n.
1, 0, 2, -3, 0, 2, 0, -13, 0, 4, 21, 0, -17, 0, 2, 0, 177, 0, -7, 0, 4, -77, 0, 2401, 0, -25, 0, 4, 0, -4987, 0, 1123, 0, -29, 0, 8, 1155, 0, -24749, 0, 1499, 0, -11, 0, 2, 0, 718657, 0, -341521, 0, 1157, 0, -74, 0, 4, -4389, 0, 6361429, 0, -495469, 0, 2411, 0, -41, 0, 4
Offset: 0
Examples
Triangle starts: [0] 1; [1] 0, 2; [2] -3, 0, 2; [3] 0, -13, 0, 4; [4] 21, 0, -17, 0, 2; [5] 0, 177, 0, -7, 0, 4; [6] -77, 0, 2401, 0, -25, 0, 4; [7] 0, -4987, 0, 1123, 0, -29, 0, 8; [8] 1155, 0, -24749, 0, 1499, 0, -11, 0, 2; [9] 0, 718657, 0, -341521, 0, 1157, 0, -74, 0, 4.
Links
- R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer, 2010. (p. 213-216.)
- Alexey Kuznetsov, Expansion of the Riemann Xi Function in Meixner-Pollaczek Polynomials. Canadian Mathematical Bulletin, 51(4) (2008), 561-569.
- Dan Romik, Orthogonal polynomial expansions for the Riemann xi function. arXiv:1902.06330 [math.NT], 2019.
Programs
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Maple
gf := ((1 - I*z)/(1 + I*z))^(I*x)*(1 + z^2)^(-3/4): serz := series(gf, z, 22): coeffz := n -> coeff(serz, z, n): row := n -> seq(numer(coeff(coeffz(n), x, k)), k = 0..n): seq(row(n), n = 0..10); # Alternative: CoeffList := p -> numer(PolynomialTools:-CoefficientList(p, x)): P := proc(n) option remember; if n = 0 then 1 elif n = 1 then 2*x else expand((1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))) fi end: ListTools:-Flatten([seq(CoeffList(P(n)), n = 0..10)]);
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Mathematica
ForceSimpl[a_] := Collect[Expand[Simplify[FunctionExpand[a]]], x] f[n_] := I^n Sum[(-1)^k Binomial[-3/4 + I*x, k] Binomial[-3/4 - I*x, n-k], {k, 0, n}] // ForceSimpl; row[n_] := CoefficientList[f[n], x] // Numerator; Table[row[n], {n, 0, 10}] // Flatten
Formula
T(n, k) = numerator([x^k] P(n, x), where P(n, x) = i^n*Sum_{k=0..n} (-1)^k* binomial(-3/4 + i*x, k)*binomial(-3/4 - i*x, n - k). The polynomials have the recurrence P(n, x) = (1/n)*(2*x*P(n - 1, x) - (n - 1/2)*P(n - 2, x))), starting with P(0, x) = 1 and P(1, x) = 2*x.