In this paper, transformation formulas for a large class of Eisenstein series defined for $${\text {Re}}(s)>2$$ Re ( s ) > 2 and $${\text {Im}}(z)>0$$ Im ( z ) > 0 by [Equation not available: see fulltext.]are investigated for $$s=1-r$$ s = 1 - r , $$r\in {\mathbb {N}}$$ r ∈ N . Here $$\left\{ f(n)\right\} $$ f ( n ) and $$\left\{ f^{*}(n)\right\} $$ f ∗ ( n ) , $$-\infty<n<\infty $$ - ∞ < n < ∞ are sequences of complex numbers with period $$k>0$$ k > 0 , and $$A_{\alpha }=\left\{ f(\alpha n)\right\} $$ A α = f ( α n ) and $$B_{\beta }=\left\{ f^{*}(\beta n)\right\} $$ B β = f ∗ ( β n ) , $$\alpha ,\beta \in {\mathbb {Z}}$$ α , β ∈ Z . Appearing in the transformation formulas are generalizations of Dedekind sums involving the periodic Bernoulli function. Reciprocity law is proved for periodic Apostol–Dedekind sum outside of the context of the transformation formulas. Furthermore, transformation formulas are presented for $$G(z,s;A_{\alpha },I;r_{1},r_{2})$$ G ( z , s ; A α , I ; r 1 , r 2 ) and $$G(z,s;I,A_{\alpha };r_{1},r_{2})$$ G ( z , s ; I , A α ; r 1 , r 2 ) , where $$I=\left\{ 1\right\} $$ I = 1 . As an application of these formulas, analogues of Ramanujan’s formula for periodic zeta-functions are derived.
Monatshefte f�r Mathematik – Springer Journals
Published: Jan 30, 2017
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