OFFSET
0,2
COMMENTS
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).
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.
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.
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
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.
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.
From Wolfdieter Lang, Dec 04 2011: (Start)
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)).
G(z,x) satisfies the ordinary first-order inhomogeneous differential equation, derived from the recurrence:
(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:
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)
Note that these are not the associated Laguerre polynomials as usually defined in the literature. - Peter Bala, Mar 22 2024
REFERENCES
M. E. H. Ismail (two chapters by W. Van Assche), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005.
FORMULA
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.
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
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
EXAMPLE
n\k 0 1 2 3 4 5 6 7
0: 1
1: -3 1
2: 11 -8 1
3: -50 58 -15 1
4: 274 -444 177 -24 1
5: -1764 3708 -2016 416 -35 1
6: 13068 -33984 23544 -6560 835 -48 1
7: -109584 341136 -288360 101560 -17370 1506 -63 1
...
MATHEMATICA
Ei[1, y_] := -ExpIntegralEi[-y];
(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 *)
PROG
(PARI)
p(n) = numerator(bestapprPade(Ser(vector(2*n, k, (k-1)!))));
concat(vector(9, n, Vec((-1)^(n-1)*p(n)))) \\ Gheorghe Coserea, Dec 01 2016
CROSSREFS
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]
KEYWORD
AUTHOR
Wolfdieter Lang, Nov 25 2011
STATUS
approved