Some families of Mathieu a-series and alternating Mathieu a-series

The main purpose of this paper is to present a number of potentially useful integral representations for the familiar Mathieu a-series as well as for its alternating version. These results are derived here from many different considerations and are shown to yield sharp bounding inequalities involving the Mathieu and alternating Mathieu a-series. Relationships of the Mathieu a-series with the Riemann Zeta function and the Dirichlet Eta function are also considered. Such special functions as the classical Bessel function J"@n(z) and the confluent hypergeometric functions "0F"1 and "1F"2 are characterized by means of certain Fredholm type integral equations of the first kind, which are associated with some of these Mathieu type series. Several integrals containing Mathieu type series are also evaluated. Finally, some closely-related new questions and open problems are indicated with a view to motivating further investigations on the subject of this paper.

[1]  T. Pogány,et al.  On integral representation of Bessel function of the first kind , 2005 .

[2]  Hari M. Srivastava,et al.  Some problems and solutions involving Mathieu's series and its generalizations. , 2004 .

[3]  O. Emersleben.  Über die Reihe $$\sum\limits_{k = 1}^\infty {\tfrac{k}{{(k^2 + c^2 )^2 }}} $$ . , 1952 .

[4]  W. Szpankowski.  Average Case Analysis of Algorithms on Sequences , 2001 .

[5]  G. Hardy,et al.  The General Theory Of Dirichlets Series , .

[6]  Tibor K. Pogány,et al.  Integral representation of Mathieu (a, λ)-series , 2005 .

[7]  Á. Elbert.  Asymptotic expansion and continued fraction for mathieu's series , 1982 .

[8]  J. Brenner,et al.  On Mathieu's Inequality , 1998 .

[9]  Hari M. Srivastava,et al.  Certain classes of series associated with the Zeta and related functions , 2003, Appl. Math. Comput..

[10]  D. S. Mitrinovic,et al.  Classical and New Inequalities in Analysis , 1992 .

[11]  T. Pogány.  Integral representation of a series which includes the Mathieu a-series , 2004 .

[12]  E. Hansen.  A Table of Series and Products , 1977 .

[13]  Kostadin Tren.  ON AN OPEN PROBLEM OF BAI-NI GUO AND FENG QI , 2003 .

[14]  Z. Tomovski.  NEW DOUBLE INEQUALITIES FOR MATHIEU TYPE SERIES , 2004 .

[15]  Hari M. Srivastava,et al.  Series Associated with the Zeta and Related Functions , 2001 .

[16]  L. Landau,et al.  Monotonicity and bounds on Bessel functions. , 2000 .

[17]  P. H. Diananda.  Some inequalities related to an inequality of Mathieu , 1980 .

[18]  I. S. Gradshteyn.  Table of Integrals, Series and Products, Corrected and Enlarged Edition , 1980 .

[19]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[20]  L. P. Kok,et al.  Table erratum: A table of series and products [Prentice-Hall, Englewood Cliffs, N.J., 1975] by E. R. Hansen , 1986 .

[21]  Bai-Ni Guo (郭白妮).  Note on Mathieu's inequality , 2000 .

[22]  P. Cerone.  Bounding Mathieu Type Series , 2003 .

[23]  Feng Qi,et al.  Inequalities for Mathieu's Series , 2000 .

[24]  Feng Qi,et al.  Integral Expression and Inequalities of Mathieu Type Series , 2003 .

[25]  Christopher T. Lenard,et al.  On Integral Forms of Generalised Mathieu Series , 2003 .

[26]  I. S. Gradshteyn,et al.  Table of Integrals, Series, and Products , 1976 .