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A136239 Forced end points ( -Infinity ->-1) integration of A137286: Triangle of coefficients of Integrated recursive orthogonal Hermite polynomials given in Hochstadt's book : P(x, n) = x*P(x, n - 1) - n*P(x, n - 2).

%I A136239 #6 May 28 2013 03:10:52
%S A136239 1,0,1,-1,0,1,-1,-3,0,1,9,0,-6,0,1,-1,27,0,-10,0,1,-19,0,65,0,-15,0,1,
%T A136239 -1,-165,0,135,0,-21,0,1,399,0,-624,0,252,0,-28,0,1,-1,2145,0,-1750,0,
%U A136239 434,0,-36,0,1
%N A136239 Forced end points ( -Infinity ->-1) integration of A137286: Triangle of coefficients of Integrated recursive orthogonal Hermite polynomials given in Hochstadt's book : P(x, n) = x*P(x, n - 1) - n*P(x, n - 2).
%C A136239 Because of error functions in the result where constants should be this is a difficult calculation.
%C A136239 Probably the wrong approach, but it is my best effort at getting Gaussian normal type functions to give integers. There has got to be a better way than this: maybe a conformal transform of the known Chebyshev Integration polynomials?
%C A136239 No recurrence formula was found for these polynomials, so they are probably wrong.
%C A136239 Row sums are:
%C A136239 {1, 1, 0, -3, 4, 17, 32, -51, 0, 793}
%D A136239 page 8 and pages 42 - 43 : Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986
%F A136239 P(x, n) = x*P(x, n - 1) - n*P(x, n - 2); L(x,n)=Integrate[Exp[y^2/4]*p(y,n-1),{y,-Infinity,x}]/(-2*Exp[ -x^2/4]) Here the weight function is taken as the square root of the Hermite weight function Exp[ -x^2/2] and then divided out of the end result.
%e A136239 {1},
%e A136239 {0, 1},
%e A136239 {-1, 0, 1},
%e A136239 {-1, -3, 0, 1},
%e A136239 {9, 0, -6, 0, 1},
%e A136239 {-1, 27, 0, -10, 0, 1},
%e A136239 {-19, 0, 65, 0, -15, 0, 1},
%e A136239 {-1, -165, 0, 135, 0, -21, 0,1},
%e A136239 {399, 0, -624, 0, 252, 0, -28, 0, 1},
%e A136239 {-1, 2145, 0, -1750, 0, 434, 0, -36, 0, 1}
%Y A136239 Cf. A137286.
%K A136239 uned,tabl,sign
%O A136239 1,8
%A A136239 _Roger L. Bagula_, Mar 16 2008