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Zbl 1195.26038
Ban, B.D.
Quotient mean series. (English)
[J] Banach J. Math. Anal. 4, No. 2, 87-99, electronic only (2010). ISSN 1735-8787/e

The quotient mean series $S^{s,t}_{p,q} $ are defined as $$ S^{s,t}_{p,q}(r_1,r_2)=\sum_{n=1}^{\infty} \frac{(M_2^{[s]}(n,r_1))^t}{(M_2^{[q]}(n,r_2))^p},$$ where $M_n^{[r]}(a_1, \dots , a_n)$ is a mean of order $r$ of an $n$-tuple $ (a_1, \dots , a_n)$. They are generalizations of the Mathieu series $S_M(r)= \sum_{i=1}^{\infty} \frac{2n}{(n^2+r^2)^2}$. The author gives an integral representation of the quotient mean series in the following form $$ S^{s,t}_{p,q}(r_1,r_2)= 2^{p/q-t/s}\,\frac{p}{q} \int_0^\infty \int_0^{[u^{1/q}]}\frac{{\bold d }_w((w^s+r_1^s)^{t/s})}{(u+r_2^q)^{p/q+1}}\,dw\, du $$ where ${\bold d}_xa(x)=a(x)+\{x\}a'(x)$. Similar representations for an alternating variant of the quotient mean series and bilateral inequalities are given. Also, special cases of quotient mean series involving Bessel functions of the first kind are considered.
[Sanja Varošanec (Zagreb)]

MSC 2000:: *26D15 Inequalities for sums, series and integrals of real functions
26E60 Means
33C10 Cylinder functions, etc.
40B05 Multiple sequences and series

Keywords: Dirichlet series; quotient mean series; Mathieu series; mean; Bessel function of the first kind

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