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Integral representation of a series which includes the Mathieu $\mathbf a$-series

Intgeral expression is deduced for the series $$\mathfrak S(r, \mu, \nu, \mathbf a) = \sum_{;n=1};^\infty \frac{;_2F_1(\frac{;\nu-\mu+1};2, \frac{;\nu-\mu};2+1, \nu+1 -\frac{;r^2};{;a(n)^2};)};{;a(n)^{;\nu-\mu+1};(a(n)^2+r^2)^{;\mu-1/2};};, $$ where $r>0, \mu>1/2, \nu+1>\mu$ and $\mathbf a:\ 0<a(1)<a(2)< \cdots <a(n) \uparrow \infty$ and $_2F_1$ is the Gauss hypergemetric function. The result precizes the intergal expression for the generalized Qi type Mathieu $\mathbf a$-series $S(r, p, \mathbf a) = \sum_{;n=1};^\infty a(n)(a(n)^2+r^2)^{;-p-1};$ given in [J.Inequal. Pure Appl. Math. 4(2003), (4.5)] generalizing some other results by Cerone and Lenard, Tomovski and Qi as well. Bounding inequalities are given for $\mathfrak S(r, \mu, \nu, \mathbf a)$ using the derived intergal expression.

Bessel function of first kind; Dirichlet series; generalized Mathieu series; Gauss hypergeometric function; Laplace integral; Mathieu $\mathbf a$-series

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