a foolish error sometimes fruitful: connection between hermitean and bessel-polynomials

[Gottfried Helms]

Seqfans -

while dealing with some derivates I made a foolish error;
but constructed from that a table of coefficients, which
occur in two paragraphs in OEIS and Mathworld:

           1*s0
           1*a*s1
       1*s1      1*a^2*s2
       3*a*s2    1*a^3*s3
   3*s2      6*a^2*s3    1*a^4*s4
  15*a*s3   10*a^3*s4    1*a^5*s5

The terms one time collected for equal powers of a and
one time collected for equal indexes of s_k give two series,
which I find in OEIS with links to hermitean polynomials
and bessel-polynomials in mathworld.

Maybe someone finds this interesting if it's not already
well known (My own table is completely useless though ;-) )
Read the hermitean coefficients below along the diagonals,
(awith one additional shift for each line)
and the bessel-polynomial-coefficients horizontal to see the
incidence.

(Sloane's A096713) // hermitean-polynomials
(Sloane's A001497) // Bessel-polynomials.

Gottfried Helms



//--------------- Hermitean polynomials (from mathworld.wolfram.com)

A modified version of the Hermite polynomial is sometimes (but rarely) defined by
He_n(x)=2^(-n/2)H_n(x/(sqrt(2)))	(55)

(Jörgensen 1916; Magnus and Oberhettinger 1948; Slater 1960,
 p. 99; Abramowitz and Stegun 1972, p. 778).
The first few of these polynomials are given by
He_1(x)	=	x	(56)
He_2(x)	=	x^2-1	(57)
He_3(x)	=	x^3-3x	(58)
He_4(x)	=	x^4-6x^2+3	(59)
He_5(x)	=	x^5-10x^3+15x.	(60)

When ordered from smallest to largest powers, the triangle
of nonzero coefficients is
 1; 1; -1, 1; -3, 1; 3, -6, 1; 15, -10, 1; ... (Sloane's A096713).



//------------- Bessel-polynomials (from mathworld.wolfram.com)

Krall and Fink (1948) defined the Bessel polynomials as the function
y_n(x)	=	sum_(k==0)^(n)((n+k)!)/((n-k)!k!)(x/2)^k	(1)
	=	sqrt(2/(pix))e^(-x)K_(-n-1/2)(1/x),	(2)

where K_n(x) is a modified Bessel function of the second kind.
They are very similar to the modified spherical bessel function
of the second kind k_n(x). The first few are
y_0(x)	=	1	(3)
y_1(x)	=	x+1	(4)
y_2(x)	=	3x^2+3x+1	(5)
y_3(x)	=	15x^3+15x^2+6x+1	(6)
y_4(x)	=	105x^4+105x^3+45x^2+10x+1	(7)

(Sloane's A001497).