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Quotient mean series. (English) Zbl 1195.26038
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.
26D15Inequalities for sums, series and integrals of real functions
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
40B05Multiple sequences and series
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