PROBABILISTIC DERIVATION OF A BILINEAR SUMMATION FORMULA FOR THE MEIXNER-POLLACZEK POLYNOMIALS

Using the technique of canonical expansion in probability theory, a bilinear summation formula is derived for the special case of the Meixner-Pollaczek polynomials {λn(k)(x)} which are defined by the generating function ∑n=0∞λn(k)(x)zn/n!=(1


INTRODUCTION
Let U be a Cauchy random variable with the probability density function (p.d.f.) () This is the hyperbolic secant distribution considered by Baten  [2], and is a special case of the generalizdd hyperbolic secant distribution treated by Harkness and Harkness I0].
Let X and X 2 be two random variables having additive random elements in comnon [6], i.e.X V + V 2 X 2 V 2 + V 3 where V. (i I, 2, 3) are mutually independent random variables each having the p.d.f, given in (I).The joint p.d.f.P(Xl, x 2) of X and X 2 is -<x2 < =.
The marginal p.d.f.'s for X and X 2 are respectively g(xl) -o P(xI' x2)dx2 2xl cosech Xl, h(x 2) _ P(Xl, x2)dx 2x 2 cosech x 2. () The orthogonal polynomials with the above marginals as weight function are related to the Euler nt,bers and have been discussed by Carlitz in [4].
Specifically, for the weight function Pk(X) . . . .sech Xl sech x2 '''sech (x Xl x2 )dx ldx2 dXk_ I, k 2, 3, ( P l(X) sech x, the polynomials I n(k) (x)} with generating function are the orthogonal polynomials in the interval (-, ) and satisfy the or thogonality condition _ Pk(X) x(k) (ix)X (k) (ix)dx (-l)nn (k) n m,n m n (6) where i /Z and denotes the Kronecker delta.m,n The explicit form of the orthogonal polynomial is given by Carlitz [4] as The last result follows easily from the following well-known generating function for the Gaussian hypergeometric function 2Fl [7, p. 82] I-n, b; c; x]z n n=0 A related system of polynomials has been discussed by Bateman who referred to them as the Mittag-Leffler polynomials.It happens that both the polynomials discussed by Bateman and Carlitz are but special cases of the system of orthogonal polynomials first discussed by Meixner Ill] and later independently by Pollaczek [12].Following the notation of [8, p. 219] (See also [5, p. 184]), the Meixner-Pollaczek polynomials are given explicitly by where a > 0, 0 < < and < x < (R).
These polynomials are orthogonal with respect to the weight function 2a- The orthogonality relation is given by tnp (a)(x; ) (l tell) -a+ix (l te-i)-a-ix, it < I.
A BILINEAR SUMMATION FORMULA From the generating function in (5) it is innediately clear that %(k) (x) n satisfies the following so-called Runge-type identity and all n.
It has been shown that the result in (I0) is both necessary and suffic'ient for the joint p.d.f, in (2) to possess a bilinear expansion (also called a canonical expansion in statistical literature) of the form [6] P(Xl, x 2) g(Xl)h(x2 where the canonical variables {en(X) } ({n(X) }) are a complete set of orthonormal polynomials with weight function g(x) (h(x)).The canonical correlation is Pn E [0n(Xl) n(X2) where E denotes the expectation operation.
For the joint p.d.f, in (2) with the equal marginal p.d.f.'s given in (3), we note that the canonical variable in this case is On(X) /n'.( 2) We therefore have the following interesting bilinear sunmation formula for the Meixner-Pollaczek polynomials

A GENERALIZATION
Consider the following more general scheme of additive random variables as in [9].
Let {i } for i l, 2, n m, {n i} for i l, 2, m and {i } for i l, 2, n 2 m where .<m < min(n l, n 2) be (n + n 2 -m) mutually independent random variables each having the p.d.f.
It is clear that the joint characteristic function (l' 2 of X and X 2 is EeiqJ] EEe i%] sech v dv sech() for < i .<n -m, .<j .<m,.<k.< n2-m.[sech(Sx + y) ]U+ldx -r(u + 1) eiVy/8 (14) The respective canonical variables are .-n By a repeated application of the Runge-type identity in (I0) analogous to the derivation leading to the result in (l|), it may be shown that the canonical correlation in this case is (m) n n l(nl) n(n2 n .<m < min(n I, n2).
On the other hand, note that from ( 14  where (S(x) denotes the Dirac delta rune=ion.
It is perhaps interesting to note in passing that a comparison of the two results in ( 12) and ( 16) allows us to deduce the following special case of the G-function, viz.ACKNOWLEDGEMENT.The author is grateful to the referee for his helpful comments, in particular regarding the connection of the system of orthogonal polynomials discussed by Carlitz with that of the Meixner-Pollaczek polynomials.