Quotient mean series (Q967187)

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 \[ S^{s,t}_{p,q}(r_1,r_2)= 2^{p/q-t/s}\,\frac{p}{q} \int_0^\infty \int_0^{[u^{1/q}]}\frac{{\mathbf d }_w((w^s+r_1^s)^{t/s})}{(u+r_2^q)^{p/q+1}}\,dw\, du \] where \({\mathbf d}_xa(x)=a(x)+\{x\}a'(x)\). 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.