TOPICS
Search

Meixner Polynomial of the Second Kind


The polynomials M_k(x;delta,eta) which form the Sheffer sequence for

g(t)={[1+deltaf(t)]^2+[f(t)]^2}^(eta/2)
(1)
f(t)=tan(t/(1+deltat))
(2)

which have generating function

 sum_(k=0)^infty(M_k(x;delta,eta))/(k!)t^k 
 =[(1+deltat)^2]^(-eta/2)exp((xtan^(-1)t)/(1-deltatan^(-1)t)).
(3)

The first few are

M_0(x;delta,eta)=1
(4)
M_1(x;delta,eta)=x-deltaeta
(5)
M_2(x;delta,eta)=x^2+2delta(1-eta)x+eta[(eta+1)delta^2-1].
(6)

See also

Meixner Polynomial of the First Kind, Sheffer Sequence

Explore with Wolfram|Alpha

References

Chihara, T. S. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach, p. 179, 1978.Roman, S. The Umbral Calculus. New York: Academic Press, 1984.

Referenced on Wolfram|Alpha

Meixner Polynomial of the Second Kind

Cite this as:

Weisstein, Eric W. "Meixner Polynomial of the Second Kind." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MeixnerPolynomialoftheSecondKind.html

Subject classifications