A class of identities associated with Dirichlet series satisfying Hecke’s functional equation

We consider two sequences a ( n stretchy="false">) a(n) and alttext="b b encoding="application/x-tex">b(n) , alttext="1 less-than-or-equal-to greater-than normal infinity"> 1 ≤ &gt; ∞ encoding="application/x-tex">1\leq n&gt;\infty generated by Dirichlet series of the forms ∑<!-- ∑ <mml:mrow class="MJX-TeXAtom-ORD"> = λ<!-- λ <mml:mi>s and μ<!-- μ <mml:mo>, encoding="application/x-tex">\begin{equation*} \sum _{n=1}^{\infty }\dfrac {a(n)}{\lambda _n^{s}}\qquad \text {and}\qquad {b(n)}{\mu _n^{s}}, \end{equation*} satisfying familiar functional equation involving gamma function alttext="normal upper Gamma mathvariant="normal">Γ<!-- Γ encoding="application/x-tex">\Gamma (s) . A general identity is established. Appearing on one side an infinite modified Bessel functions alttext="upper K nu"> K ν<!-- ν </mml:msub> encoding="application/x-tex">K_{\nu } wherein other that analogue Hurwitz zeta function. Six special cases, including tau τ<!-- τ encoding="application/x-tex">a(n)=\tau (n) r k r k encoding="application/x-tex">a(n)=r_k(n) are examined, where alttext="tau encoding="application/x-tex">\tau Ramanujan’s arithmetical alttext="r encoding="application/x-tex">r_k(n) denotes number representations alttext="n"> encoding="application/x-tex">n as sum alttext="k"> encoding="application/x-tex">k squares. All but examples appear to be new.