On the mean values of Dedekind sum and certain Hardy–Berndt sums

This paper investigates the mean value behavior of Apostol-type Dedekind and certain Hardy–Berndt sums when evaluated along sequences defined by second-order linear recurrences. By employing reciprocity formulas together with the structural properties of these sequences, we derive explicit and asymptotic formulas for a broad class of sums, including <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> </m:mfenced> </m:math> sₚ(a,c) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> </m:msub> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> </m:mfenced> </m:math> Sₚ(a,c) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:mfenced> </m:mrow> </m:msubsup> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> </m:mfenced> </m:math> Sₚ⁽³⁾(a,c) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msubsup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mfenced close=")" open="("> <m:mrow> <m:mn>5</m:mn> </m:mrow> </m:mfenced> </m:mrow> </m:msubsup> <m:mfenced close=")" open="("> <m:mrow> <m:mi>a</m:mi> <m:mo>,</m:mo> <m:mi>c</m:mi> </m:mrow> </m:mfenced> </m:math> Sₚ⁽⁵⁾(a,c) . In particular, we establish the unboundedness of these sums in a generalized setting and show that both their pointwise values and associated mean values exhibit a relative asymptotic growth behavior that is essentially independent of the parameter p , revealing a certain universality phenomenon.