an: Zbl 1195.26038
au: Ban, B.D.
ti: Quotient mean series.
la: EN
so: Banach J. Math. Anal. 4, No. 2, 87-99, electronic only (2010).
py: 2010
dt: J
cc: *26D15 26E60 33C10 40B05
ut: Dirichlet series; quotient mean series; Mathieu series; mean; Bessel
    function of the first kind
ab: 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.
rv: Sanja Varo\v sanec (Zagreb)