Möbius inversion formulas related to the Fourier expansions of two-dimensional Apostol–Bernoulli polynomials

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The two-dimensional (2D) Apostol–Bernoulli and Apostol–Euler polynomials are defined via the generating functionstext+ytmλet−1=∑n=0∞Bn(x,y;λ)tnn!,2ext+ytmλet+1=∑n=0∞En(x,y;λ)tnn!. The Apostol–Bernoulli and Apostol–Euler polynomials are essentially the same as parametrized polynomial families, thus we may restrict to the latter.The Fourier coefficients of x↦λxBn(x,y;λ) on [0,1) satisfy an arithmetical–dynamical transformation formula which makes the Fourier series amenable to a technique of generalized Möbius inversion. This yields some interesting arithmetic summation identities, among them parametrized versions of the following well-known classical formula of Davenport:∑k=1∞μ(k)k{kx}=−sin⁡(2πx)π(x∈R), where μ(n) is the Möbius function and {x} denotes the fractional part of x. Davenport's formula is the limiting case α=0 of−4π4π2−α2sin⁡(2πx)=∑k=1∞μ(k)k⋅sin⁡(αk({kx}−12))2sin⁡(α2k), which is valid for −π<α≤π.