Meixner–Pollaczek polynomials

In mathematics, the Meixner–Pollaczek polynomials are a family of orthogonal polynomials P(λ)
n(x,φ) introduced by Meixner (1934), which up to elementary changes of variables are the same as the Pollaczek polynomials Pλ
n(x,a,b) rediscovered by Pollaczek (1949) in the case λ=1/2, and later generalized by him.

They are defined by

P n ( λ ) ( x ; ϕ ) = ( 2 λ ) n n ! e i n ϕ 2 F 1 ( − n ,   λ + i x 2 λ ; 1 − e − 2 i ϕ ) {\displaystyle P_{n}^{(\lambda )}(x;\phi )={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}\left({\begin{array}{c}-n,~\lambda +ix\\2\lambda \end{array}};1-e^{-2i\phi }\right)}
P n λ ( cos ⁡ ϕ ; a , b ) = ( 2 λ ) n n ! e i n ϕ 2 F 1 ( − n ,   λ + i ( a cos ⁡ ϕ + b ) / sin ⁡ ϕ 2 λ ; 1 − e − 2 i ϕ ) {\displaystyle P_{n}^{\lambda }(\cos \phi ;a,b)={\frac {(2\lambda )_{n}}{n!}}e^{in\phi }{}_{2}F_{1}\left({\begin{array}{c}-n,~\lambda +i(a\cos \phi +b)/\sin \phi \\2\lambda \end{array}};1-e^{-2i\phi }\right)}

Examples

The first few Meixner–Pollaczek polynomials are

P 0 ( λ ) ( x ; ϕ ) = 1 {\displaystyle P_{0}^{(\lambda )}(x;\phi )=1}
P 1 ( λ ) ( x ; ϕ ) = 2 ( λ cos ⁡ ϕ + x sin ⁡ ϕ ) {\displaystyle P_{1}^{(\lambda )}(x;\phi )=2(\lambda \cos \phi +x\sin \phi )}
P 2 ( λ ) ( x ; ϕ ) = x 2 + λ 2 + ( λ 2 + λ − x 2 ) cos ⁡ ( 2 ϕ ) + ( 1 + 2 λ ) x sin ⁡ ( 2 ϕ ) . {\displaystyle P_{2}^{(\lambda )}(x;\phi )=x^{2}+\lambda ^{2}+(\lambda ^{2}+\lambda -x^{2})\cos(2\phi )+(1+2\lambda )x\sin(2\phi ).}

Properties

Orthogonality

The Meixner–Pollaczek polynomials Pm(λ)(x;φ) are orthogonal on the real line with respect to the weight function

w ( x ; λ , ϕ ) = | Γ ( λ + i x ) | 2 e ( 2 ϕ − π ) x {\displaystyle w(x;\lambda ,\phi )=|\Gamma (\lambda +ix)|^{2}e^{(2\phi -\pi )x}}

and the orthogonality relation is given by[1]

∫ − ∞ ∞ P n ( λ ) ( x ; ϕ ) P m ( λ ) ( x ; ϕ ) w ( x ; λ , ϕ ) d x = 2 π Γ ( n + 2 λ ) ( 2 sin ⁡ ϕ ) 2 λ n ! δ m n , λ > 0 , 0 < ϕ < π . {\displaystyle \int _{-\infty }^{\infty }P_{n}^{(\lambda )}(x;\phi )P_{m}^{(\lambda )}(x;\phi )w(x;\lambda ,\phi )dx={\frac {2\pi \Gamma (n+2\lambda )}{(2\sin \phi )^{2\lambda }n!}}\delta _{mn},\quad \lambda >0,\quad 0<\phi <\pi .}

Recurrence relation

The sequence of Meixner–Pollaczek polynomials satisfies the recurrence relation[2]

( n + 1 ) P n + 1 ( λ ) ( x ; ϕ ) = 2 ( x sin ⁡ ϕ + ( n + λ ) cos ⁡ ϕ ) P n ( λ ) ( x ; ϕ ) − ( n + 2 λ − 1 ) P n − 1 ( x ; ϕ ) . {\displaystyle (n+1)P_{n+1}^{(\lambda )}(x;\phi )=2{\bigl (}x\sin \phi +(n+\lambda )\cos \phi {\bigr )}P_{n}^{(\lambda )}(x;\phi )-(n+2\lambda -1)P_{n-1}(x;\phi ).}

Rodrigues formula

The Meixner–Pollaczek polynomials are given by the Rodrigues-like formula[3]

P n ( λ ) ( x ; ϕ ) = ( − 1 ) n n ! w ( x ; λ , ϕ ) d n d x n w ( x ; λ + 1 2 n , ϕ ) , {\displaystyle P_{n}^{(\lambda )}(x;\phi )={\frac {(-1)^{n}}{n!\,w(x;\lambda ,\phi )}}{\frac {d^{n}}{dx^{n}}}w\left(x;\lambda +{\tfrac {1}{2}}n,\phi \right),}

where w(x;λ,φ) is the weight function given above.

Generating function

The Meixner–Pollaczek polynomials have the generating function[4]

∑ n = 0 ∞ t n P n ( λ ) ( x ; ϕ ) = ( 1 − e i ϕ t ) − λ + i x ( 1 − e − i ϕ t ) − λ − i x . {\displaystyle \sum _{n=0}^{\infty }t^{n}P_{n}^{(\lambda )}(x;\phi )=(1-e^{i\phi }t)^{-\lambda +ix}(1-e^{-i\phi }t)^{-\lambda -ix}.}

See also

  • Sieved Pollaczek polynomials

References

  1. ^ Koekoek, Lesky, & Swarttouw (2010), p. 213.
  2. ^ Koekoek, Lesky, & Swarttouw (2010), p. 213.
  3. ^ Koekoek, Lesky, & Swarttouw (2010), p. 214.
  4. ^ Koekoek, Lesky, & Swarttouw (2010), p. 215.
  • Koekoek, Roelof; Lesky, Peter A.; Swarttouw, René F. (2010), Hypergeometric orthogonal polynomials and their q-analogues, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-642-05014-5, ISBN 978-3-642-05013-8, MR 2656096
  • Koornwinder, Tom H.; Wong, Roderick S. C.; Koekoek, Roelof; Swarttouw, René F. (2010), "Pollaczek Polynomials", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248.
  • Meixner, J. (1934), "Orthogonale Polynomsysteme Mit Einer Besonderen Gestalt Der Erzeugenden Funktion", J. London Math. Soc., s1-9: 6–13, doi:10.1112/jlms/s1-9.1.6
  • Pollaczek, Félix (1949), "Sur une généralisation des polynomes de Legendre", Les Comptes rendus de l'Académie des sciences, 228: 1363–1365, MR 0030037