A PROBABILISTIC PROOF OF THE SERIES REPRESENTATION OF THE MACDONALD FUNCTION WITH APPLICATIONS

. A series representation of the Macdonald function is obtained using the properties of a probability density function and its moment generating function. Some applications of the result are discussed and an open problem is posed.

1. INTRODUCTION.Kadell [S] used the prohabilistic approach to prove Ramanujan's ,, sum.Ismail [6] gave the most natural proof of Ramanujan 1bl sum that extended to multivariate hypergeometric functions.Some algebraic and other techniques are used to provide simple proofs of the established identities, see [1, 4, 5, 6, 8, 9].In this paper we have used probabilistic approach to derive a series representation [2, p. 100] of the Macdonald function.Some applications of the result are discussed and an open problem is posed.xp(--)e, -< , < , > 0.
Then, the function defined by From (2.7) and (2.8) we get I(a,(1 t)) The substitution z in (2.9) yields the proof of the theorem.
In particular when 0 and z in (2.9) we get K0(2z/il -t)) _, {z"K,.,(2z)}n' is the probability density function (pdf).It may be noted that the pdf (2.3) has appeared in an earlier work.This is the limiting case a 0 of theorem 1.11 in [7].
The r-th non-central moment of the random variable X having (2.3) as its pdf is given by E(X) x"f(x)dx I(a + r, fl) (2.4) I(oe,) The moment generating function (mgf) of f(x) is given by x -' exp(-(1 t)x x-)dx, 0 _< < 1. (2.5)

S(et)=_,S(X').
r--O yields, I(a,(1 t)) I(( + r,f) t" (1-t)'I(a,B) ,.=o I(a,) r! (2.3) which shows that Ko(zlv/'i-X-t) is the generating function of .(z/2)'Kn(z),n 0,1,2,3,....An immediate consequence of the theorem is the following result which provides the closed form solution to the representation of the first derivative with respect to the order of the Macdonald function at the integral values of the order.The problem remains open for the higher derivatives and the other values of the order of the function.
In particular when n 2 we get exp(-cosh )sinh()e [K() + K()], ( which do not sm to be known in the literature.
We state here an open problem the solution to which will have Nr-rching eonsequenc in the genereliation of the inverse Geussian distribution.
STATEMENT OF THE OPEN PROBLEM.Find the relationship of with the ther Nnctions for n > 2.
ACKNOWLEDGEMENTS.Private corrpondence with Professor M. hmen at Carleton Univer- sitN Ottewa, Canada is appreciate.The authors are indebted to the the refer for helpNl comments and to the King Fahd University of Petroleum and MinerMs for the excellent research Ndlities.