On the w−Multiple Meixner Polynomials

In this thesis, a new family of discrete MOPs, namely ω-multiple Meixner polynomials, where ω is a positive real number is introduced. For ω-MOPs, orthogonality conditions w.r.t r (with r > 1) different Pascal distributions (Negative Binomial distributions) are used. Depending on the selection of the parameters in the Negative Binomial distribution, two kinds of ω-MMPs, namely 1st and 2nd kinds are considered. Some structural properties of ω-MMPs, such as raising operator, Rodrigue’s type formula and explicit representation are derived. The generating function for ω-MMPs is obtained and by use of this generating function several consequences for these polynomials are reached. A lowering operator for ω-MMPs which will be helpful for obtaining difference equation is also derived. By combining the lowering operator with the raising operator the difference equation which has the ω-MMPs as a solution are obtained. A third order difference equation for ω-MMPs is given . Also it is shown that for the special case ω = 1, the obtained results coincide with the existing results for MMPs of both kinds. In the last part as an illustrated example for the ω-MMPs of the first kind the special case when ω = 1/2 is considered and for the 1/2-MMPs of the first kind,the results obtained for the main theorems are stated. For the ω-MMPs of the second kind the special case when ω = 5/3 is studied and for the 5/3-MMPs of the second kind, the corresponding result obtained for the main theorems are examined.