SOME FAMILIES OF MATHIEU TYPE SERIES AND HURWITZ-LERCH ZETA FUNCTIONS AND ASSOCIATED PROBABILITY DISTRIBUTIONS

By making use of the familiar Mathieu series and its generalizations, and the Riemann, Hurwitz (or generalized) and the Hurwitz-Lerch Zeta functions and their multi-parameter extensions, the authors present a systematic study of probability density functions and probability distributions associated with some generalizations of the Mathieu series and the Planck’s law. In particular, the characteristic functions and fractional moments related to the probability density functions of the considered probability distributions are derived. Integral representations of trigonometric Mathieu series, harmonic Mathieu series and some other particular forms of the Mathieu series are also given. Finally, various interesting results are proved for the Fourier- Mathieu series (which are introduced in this paper) and for their nth partial sums by applying some known theorems and lemmas for trigonometric series given by (for example) Telyakovskii, Ul’yanov, Sidon and Fomin, Bojanic and Stanojevi´c, and others.