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A199577 Coefficient triangle of the associated Laguerre polynomials of order 1. 4
1, -3, 1, 11, -8, 1, -50, 58, -15, 1, 274, -444, 177, -24, 1, -1764, 3708, -2016, 416, -35, 1, 13068, -33984, 23544, -6560, 835, -48, 1, -109584, 341136, -288360, 101560, -17370, 1506, -63, 1, 1026576, -3733920, 3736440, -1595040, 343410, -39900, 2513, -80, 1 (list; table; graph; refs; listen; history; text; internal format)
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}, 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)=int(exp(-t)/t,t=y..infty). From this the e.g.f. gLa(z,x), given in the Formula section, results.

(End)

REFERENCES

M. E. H. Ismail (two chapters by W. Van Assche), Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005.

LINKS

Table of n, a(n) for n=0..44.

FORMULA

a(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{1;n-1}(x) - (n^2)*La_{1;n-2}, 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

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

...

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]

Sequence in context: A110440 A135574 A008969 * A228534 A119908 A153257

Adjacent sequences:  A199574 A199575 A199576 * A199578 A199579 A199580

KEYWORD

sign,easy,tabl

AUTHOR

Wolfdieter Lang, Nov 25 2011

STATUS

approved

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Last modified October 18 18:10 EDT 2018. Contains 316323 sequences. (Running on oeis4.)