We solve a problem that has remained unsolved since 1936 – the exact distribution of the product of two correlated normal random variables. As a by-product, we derive the exact distribution of the mean of the product of correlated normal random variables.

This article presents a systematic investigation of various integrals and computational representations for some families of generalized Hurwitz–Lerch Zeta functions which are introduced here. We first establish their relationship with the -function, which enables us to derive the Mellin–Barnes type integral representations for nearly all of the generalized and specialized Hurwitz–Lerch Zeta functions. The integral expressions studied in this paper provide extensions of the corresponding results given by many authors, including (for example) Garg et al. [A further study of general Hurwitz-Lerch zeta function, Algebras Groups Geom. 25 (2008), pp. 311–319] and Lin and Srivastava [Some families of the Hurwitz-Lerch Zeta functions and associated fractional derivative and other integral representations, Appl. Math. Comput. 154 (2004), pp. 725–733]. We also derive a further analytic continuation formula which provides an elegant extension of the well-known analytic continuation formula for the Gauss hypergeometric function. Fractional derivatives associated with the generalized Hurwitz–Lerch Zeta functions are obtained. The relationship between the generalized Hurwitz–Lerch Zeta function and the -function, which was given by Garg et al., is seen to be erroneous and we give its corrected version here. Finally, a unification and extension of the Hurwitz–Lerch Zeta function, introduced in this article, is presented and two of its interesting special cases associated with the Mittag–Leffler type functions due to Barnes [The asymptotic expansion of integral functions defined by Taylor series, Philos. Trans. Roy. Soc. London Ser. A 206 (1906), pp. 249–297] and the generalized M-series considered recently by Sharma and Jain [A note on a generalzed M-series as a special function of fractional calculus, Fract. Calc. Appl. Anal. 12 (2009), pp. 449–452.] are deduced.

Free and self‐excited vibrations of conservative oscillators with polynomial nonlinearity are considered. Mathematical model of the system is a second‐order differential equation with a nonlinearity of polynomial type, whose terms are of integer and/or noninteger order. For the case when only one nonlinear term exists, the exact analytical solution of the differential equation is determined as a cosine‐Ateb function. Based on this solution, the asymptotic averaging procedure for solving the perturbed strong non‐linear differential equation is developed. The method does not require the existence of the small parameter in the system. Special attention is given to the case when the dominant term is a linear one and to the case when it is of any non‐linear order. Exact solutions of the averaged differential equations of motion are obtained. The obtained results are compared with “exact” numerical solutions and previously obtained analytical approximate ones. Advantages and disadvantages of the suggested procedure are discussed.

Using the Bernstein theorem we prove the complete monotonicity of the three parameter Mittag?Leffler function E??,? (?w) for w ? 0 and suitably constrained parameters ?, ? and ?.