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A199580
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Coefficient array for the monic X_1-Laguerre polynomials with parameter k=1.
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0
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0, 2, 1, -3, 0, 1, 8, -4, -4, 1, -30, 30, 15, -10, 1, 144, -216, -48, 84, -18, 1, -840, 1680, 0, -700, 245, -28, 1, 5760, -14400, 2880, 6000, -3120, 552, -40, 1, -45360, 136080, -52920, -52920, 39690, -9702, 1071, -54, 1, 403200, -1411200, 806400, 470400
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OFFSET
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0,2
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COMMENTS
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The offset is actually n>=1, the entry a(0,0)=0 has been added to have the tabl (triangle) format.
The general X_1-Laguerre orthogonal and complete poly-
nomial system (OPS) Lhat(k;n,x), with k>0 (not >-1 like in the classical Laguerre case) and n>=1 (not n>=0) has been found by Gomez-Ullate et al. (see the reference and the link), and their notation is
{L hat^{(k)}_i(x)}_{i=1}^{infinity}. Because of the start with degree n=1 they are called 1-OPS and this explains also the index 1 at X (for exceptional).
The orthogonality interval is [0,infinity) (like in the classical Laguerre case), and the (positive) weight function is What(k;x) = x^k*exp(-x)/(x+k)^2. For the second order differential equation (not of the hypergeometric type), the Rodrigues-type formula, the relation to ordinary generalized Laguerre polynomials, and the three term recurrence relation see the Gomez-Ullate et al. reference or link, eqs. (21) with (24), (77) with (16) and (76), (80) or (82), and (87) (with the z in the second term an x), respectively.
Here the monic version of the X_1-Laguerre OPS is used: mLhat(k;n,x) = ((-1)^n)*(n-1)!*Lhat(k;n,x) (not ((-1)^n)*n! like in the classical Laguerre case). For this number triangle k=1. From eq.(87) of the given reference follows the recurrence for the monic polynomials:
mLhat(k;n,x) = ((n-2+k)*((x-2*n+3-k)*(x+k)^2 + 2*k)* mLhat(k;n-1,x) - (n-2)*(n-3+k)*((n-1+k)*(x+k)^2-k)*
mLhat(k;n-2,x))/((n-2+k)*(x+k)^2-k) for n>=3, with the inputs mLhat(k;1,x)=x+k+1 and mLhat(k;2,x)= x^2-2*k - k^2.
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REFERENCES
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David Gomez-Ullate, Niky Kamran, Robert Milson, An extended class of orthogonal polynomials defined by a Sturm-Liouville problem, J. Math. Anal. Appl. (2009), 352-367.
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LINKS
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FORMULA
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a(n,m)=[x^m]mLhat(1;n,x), n>=1, m=0,...,n, with the monic orthogonal X_1-Laguerre polynomials mLhat(1;n,x) defined from the non-monic version introduced by Gomez-Ullate et al., and explained in the comment section.
Recurrence for the monic polynomials (from eq.(87), with z=x, and k=1 of the Gomez-Ullate et al. reference):
mLhat(1;n,x) = ((n-1)*((x-2*n+2)*(x+1)^2 + 2)* mLhat(1;n-1,x) - (n-2)^2*(n*(x+1)^2-1)*mLhat(1;n-2,x))/((n-1)*(x+1)^2-1)for n>=3, with the inputs mLhat(k;1,x)=x+2 and mLhat(k;2,x)= x^2-3.
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EXAMPLE
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The triangle (without the n=0 entry 0) starts:
n\m 0 1 2 3 4 5 6 7 8
1: 2 1
2: -3 0 1
3: 8 -4 -4 1
4: -30 30 15 -10 1
5: 144 -216 -48 84 -18 1
6: -840 1680 0 -700 245 -28 1
7: 5760 -14400 2880 6000 -3120 552 -40 1
8: -45360 136080 -52920 -52920 39690 -9702 1071 -54 1
...
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CROSSREFS
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Cf. A021009(n)*(-1)^n (monic Laguerre with parameter 0).
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KEYWORD
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AUTHOR
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STATUS
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approved
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