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Bessel Polynomial


BesselPolynomialY

Krall and Fink (1949) 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^(1/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)

(OEIS A001497). These functions satisfy the differential equation

 x^2y^('')+(2x+2)y^'-n(n+1)y=0.
(8)
BesselPolynomialP

Carlitz (1957) subsequently considered the related polynomials

 p_n(x)=x^ny_(n-1)(1/x).
(9)

This polynomial forms an associated Sheffer sequence with

 f(t)=t-1/2t^2.
(10)

This gives the generating function

 sum_(k=0)^infty(p_k(x))/(k!)t^k=e^(x(1-sqrt(1-2t))).
(11)

The explicit formula is

p_n(x)=sum_(k=1)^(n)((2n-k-1)!)/(2^(n-k)(k-1)!(n-k)!)x^k
(12)
=(2n-3)!!x_1F_1(1-n;2-2n;2x),
(13)

where x!! is a double factorial and _1F_1(a;b;z) is a confluent hypergeometric function of the first kind. The first few polynomials are

p_1(x)=x
(14)
p_2(x)=x^2+x
(15)
p_3(x)=x^3+3x^2+3x
(16)
p_4(x)=x^4+6x^3+15x^2+15x
(17)

(OEIS A104548).

The polynomials satisfy the recurrence formula

 p_n^('')(x)-2p_n^'(x)+2np_(n-1)(x)=0.
(18)

See also

Bessel Function, Modified Spherical Bessel Function of the Second Kind, Sheffer Sequence

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References

Carlitz, L. "A Note on the Bessel Polynomials." Duke Math. J. 24, 151-162, 1957.Grosswald, E. Bessel Polynomials. New York: Springer-Verlag, 1978.Krall, H. L. and Fink, O. "A New Class of Orthogonal Polynomials: The Bessel Polynomials." Trans. Amer. Math. Soc. 65, 100-115, 1949.Roman, S. "The Bessel Polynomials." §4.1.7 in The Umbral Calculus. New York: Academic Press, pp. 78-82, 1984.Sloane, N. J. A. Sequences A001497, A001498, and A104548 in "The On-Line Encyclopedia of Integer Sequences."

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Bessel Polynomial

Cite this as:

Weisstein, Eric W. "Bessel Polynomial." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BesselPolynomial.html

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