arXiv:1010.3348v3 [math.CA] 9 Feb 2011 File: Marcum.tex, printed: 2011-02-010, 2.08 THE GENERALIZED MARCUM Q−FUNCTION: AN ORTHOGONAL POLYNOMIAL APPROACH SZIL ´ ARD ANDR ´ AS, ´ ARP ´ AD BARICZ, AND YIN SUN Abstract. A novel power series representation of the generalized Marcum Q-function of pos- itive order involving generalized Laguerre polynomials is presented. The absolute convergence of the proposed power series expansion is showed, together with a convergence speed analysis by means of truncation error. A brief review of related studies and some numerical results are also provided. 1. Introduction For ν real number let I ν be denotes the modified Bessel function [49, p. 77] of the first kind of order ν, defined by (1.1) I ν (t)=  n≥0 (t/2) 2n+ν n!Γ(ν + n + 1) , and let b → Q ν (a, b) be the generalized Marcum Q−function, defined by (1.2) Q ν (a, b)= 1 a ν-1  ∞ b t ν e - t 2 +a 2 2 I ν-1 (at)dt, where b ≥ 0 and a, ν > 0. Here Γ stands for the well-known Euler gamma function. When ν =1, the function b → Q 1 (a, b)=  ∞ b te - t 2 +a 2 2 I 0 (at)dt is known in literature as the (first order) Marcum Q−function. The Marcum Q−function and its generalization are frequently used in the detection theories for radar systems [27] and wire- less communications [12, 13], and have important applications in error performance analysis of digital communication problems dealing with partially coherent, differentially coherent, and non-coherent detections [38, 40]. Since, the precise computations of the Marcum Q−function and generalized Marcum Q−function are quite difficult, in the last few decades several authors worked on precise and stable numerical calculation algorithms for the functions. See the papers of Dillard [14], Cantrell [7], Cantrell and Ojha [8], Shnidman [34], Helstrom [17], Temme [46] and the references therein. Moreover, many tight lower and upper bounds for the Marcum Q−function and generalized Marcum Q−function were proposed as simpler alternative evalu- ating methods or intermediate results for further integrations. See, for example, the papers of Simon [35], Chiani [10], Simon and Alouini [37], Annamalai and Tellambura [1], Corazza and Ferrari [11], Li and Kam [22], Baricz [4], Baricz and Sun [5, 6], Kapinas et al. [19], Sun et al. [41], Li et al. [23] and the references therein. In this field, the order ν is usually the number of independent samples of the output of a square-law detector, and hence in most of the papers the authors deduce lower and upper bounds for the generalized Marcum Q−function with or- der ν integer. On the other hand, based on the papers [8, 27, 34] there are introduced in the Matlab 6.5 software the Marcum Q−function and positive integer order generalized Marcum Q−function 1 : marcumq(a,b) computes the value of the first order Marcum Q−function Q 1 (a, b) and marcumq(a,b,m) computes the value of the mth order generalized Marcum Q−function Q m (a, b), defined by (1.2), where m is a positive integer. However, in some important appli- cations, the order ν> 0 of the generalized Marcum Q−function is not necessarily an integer Key words and phrases. Marcum Q-function, Laguerre polynomials, modified Bessel functions. 1 See http://www.mathworks.com/access/helpdesk/help/toolbox/signal/marcumq.html for more details. 1