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%I #45 Mar 24 2024 12:41:43
%S 1,-3,1,11,-8,1,-50,58,-15,1,274,-444,177,-24,1,-1764,3708,-2016,416,
%T -35,1,13068,-33984,23544,-6560,835,-48,1,-109584,341136,-288360,
%U 101560,-17370,1506,-63,1,1026576,-3733920,3736440,-1595040,343410,-39900,2513,-80,1
%N Coefficient triangle of the associated Laguerre polynomials of order 1.
%C Associated monic polynomials of order m (a nonnegative integer) in one variable, p_n(m;x), have the coefficients of the three-term recurrence of the original orthogonal monic polynomials p_n(x)=p_n(0;x) shifted by +m, and use the same inputs for n=-1 and n=0, namely 0 and 1, respectively. See, e.g., the Ismail reference, p. 27, Definition (2.3.4), where the notation is P_n(x;c) = p_n(c;x).
%C p_n(x)=p_n(0;x) and p_{n-1}(1;x) provide the fundamental system for the three-term recurrence of p_n(x) with general input.
%C p_{n-1}(1;x)/p_n(0;x) is the n-th approximation to the Jacobi continued fraction related to the three-term recurrence.
%C The monic row polynomials are La_n(1;x) = Sum_{k=0..n} a(n,k)*x^k, with the monic Laguerre polynomials La_n(x), which have the three-term recurrence
%C La_n(x) = (x - (2*n - 1))*La_{n-1}(x) - (n - 1)^2*La_{n-2}(x), La_{-1}(x) = 0, La_0(x) = 1.
%C In the Ismail reference the non-monic associated Laguerre polynomials of order 1 appear on p. 160 in Theorem 5.6.1, eq. 5.6.11. The connection is: La_n(1;x)= L_n^{(alpha=0)}(x;1)*(n+1)!*(-1)^n.
%C From _Wolfdieter Lang_, Dec 04 2011: (Start)
%C The e.g.f. gLa(z,x) for La_n(1;x) can be obtained from the o.g.f. G(z,x) for the non-monic version L_n^{(alpha=0)}(x;1) by gLa(z,x)=(d/dz)(z*G(-z,x)).
%C G(z,x) satisfies the ordinary first-order inhomogeneous differential equation, derived from the recurrence:
%C (d/dz)G(z,x) = (2/(1-z)+(1-x)/(1-z)^2-1/(z*(1-z)^2))* G(z,x) + 1/(z*(1-z)^2), with G(0,x)=1. The standard solution is:
%C G(z,x) = exp(-x/(1-z))*(Ei(1,-x) - Ei(1,-x/(1-z)))/(z*(1-z)), with the exponential integral Ei(1,y) = Integral_{t = y..oo} exp(-t)/t. From this the e.g.f. gLa(z,x), given in the Formula section, results. (End)
%C Note that these are not the associated Laguerre polynomials as usually defined in the literature. - _Peter Bala_, Mar 22 2024
%D M. E. H. Ismail (two chapters by W. Van Assche), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005.
%F T(n,k) = [x^k] La_n(1; x), n >= 0, k = 0,...,n, with the three-term recurrence of the row polynomials: La_n(1; x) = (x - (2*n + 1))*La_{n-1}(1; x) - (n^2)* La_{n-2}(1; x), with La_{-1}(1; x) = 0, La_0(1; x) = 1.
%F The e.g.f. for La_n(1; x) is (1 - exp(-x/(1+z))*(1-x/(1+z))*(Ei(1,-x/(1+z)) - Ei(1,-x)))/(1+z)^2, with the exponential integral Ei. See the comments section for the definition and the proof. - _Wolfdieter Lang_, Dec 04 2011
%F n-th row polynomial = (-1)^n*(n+1)!*L(n+1, x)*Sum_{k = 0..n+1} 1/(k*L(k, x)*L(k-1, x)), where L(k, x) denotes the k-th Laguerre polynomial. See A021009. - _Peter Bala_, Mar 22 2024
%e n\k 0 1 2 3 4 5 6 7
%e 0: 1
%e 1: -3 1
%e 2: 11 -8 1
%e 3: -50 58 -15 1
%e 4: 274 -444 177 -24 1
%e 5: -1764 3708 -2016 416 -35 1
%e 6: 13068 -33984 23544 -6560 835 -48 1
%e 7: -109584 341136 -288360 101560 -17370 1506 -63 1
%e ...
%t Ei[1, y_] := -ExpIntegralEi[-y];
%t (cc = CoefficientList[#, x]; cc (Length[cc]-1)!)& /@ CoefficientList[(1 - Exp[-x/(1+z)] (1 - x/(1+z)) (Ei[1, -x/(1+z)] - Ei[1, -x]))/(1+z)^2 + O[z]^9, z] // Flatten (* _Jean-François Alcover_, Oct 28 2019 *)
%o (PARI)
%o p(n) = numerator(bestapprPade(Ser(vector(2*n, k, (k-1)!))));
%o concat(vector(9, n, Vec((-1)^(n-1)*p(n)))) \\ _Gheorghe Coserea_, Dec 01 2016
%Y Cf. A021009 (Laguerre), A199578 (row sums), A002793(n+1)*(-1)^n (alternating row sums, conjecture). [This conjecture has been proved by _Wolfdieter Lang_, Dec 12 2011]
%K sign,easy,tabl
%O 0,2
%A _Wolfdieter Lang_, Nov 25 2011
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