Generalized Meixner-Pollaczek polynomials

We consider the generalized Meixner-Pollaczek (GMP) polynomials P-n(lambda)(x; theta, psi) of a variable x is an element of R and parameters lambda > 0, theta is an element of (0, pi), psi is an element of R, defined via the generating function G(lambda)(x; theta, psi; z) = 1/(1 - ze(i theta))(lambda-ix)(1-ze(i psi))(lambda+ix) = Sigma P-infinity(n=0)n(lambda)(x; theta, psi)z(n), vertical bar z vertical bar < 1. We find the three-term recurrence relation, the explicite formula, the hypergeometric representation, the difference equation and the orthogonality relation for GMP polynomials P-n(lambda) (x; theta, psi). Moreover, we study the special case of P-n(lambda) (x; theta, psi) corresponding to the choice psi = pi + theta and psi = pi - theta, which leads to some interesting families of polynomials. The limiting case (lambda -> 0) of the sequences of polynomials P-n(lambda) (x; theta, pi + theta) is obtained, and the orthogonality relation in the strip S = {z is an element of C : |F(z)| < 1} is shown.