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

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• 作者：Abdelmejid Bayad^a ; ^{abayad@maths.univ-evry.fr" class="auth_mail" title="E-mail the corresponding author} ; Luis Navas^b ; ¹ ; ^{navas@usal.es" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Davenport's formula ; Apostol&ndash ; Bernoulli polynomials ; Mö ; bius inversion ; Fourier series
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：163
• 期：Complete
• 页码：457-473
• 全文大小：383 K

文摘

The two-dimensional (2D) Apostol–Bernoulli and Apostol–Euler polynomials are defined via the generating functions 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 1600038X&_mathId=si2.gif&_user=111111111&_pii=S0022314X1600038X&_rdoc=1&_issn=0022314X&md5=5d3cd8ef7a521e3c1eeeea3c00f2b2a0" title="Click to view the MathML source">x↦λ^xB_n(x,y;λ)$x\mapsto {\lambda }^x{\mathcal{B}}_n(x,y;\lambda )$ on 1600038X&_mathId=si3.gif&_user=111111111&_pii=S0022314X1600038X&_rdoc=1&_issn=0022314X&md5=1e3912152bd1a6f6c2b64fb53ce5c5f1" title="Click to view the MathML source">[0,1)$[0,1)$ 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:

where 1600038X&_mathId=si5.gif&_user=111111111&_pii=S0022314X1600038X&_rdoc=1&_issn=0022314X&md5=c056198c0edde47c96beb1ed2ec52a8b" title="Click to view the MathML source">μ(n)$\mu (n)$ is the Möbius function and 1600038X&_mathId=si135.gif&_user=111111111&_pii=S0022314X1600038X&_rdoc=1&_issn=0022314X&md5=38306763cab809bb297b3833a5ae067d" title="Click to view the MathML source">{x}$\{x\}$ denotes the fractional part of x  . Davenport's formula is the limiting case 1600038X&_mathId=si242.gif&_user=111111111&_pii=S0022314X1600038X&_rdoc=1&_issn=0022314X&md5=13815b59d58fae9e9dc2d0c291cc4766" title="Click to view the MathML source">α=0$\alpha =0$ of which is valid for 1600038X&_mathId=si249.gif&_user=111111111&_pii=S0022314X1600038X&_rdoc=1&_issn=0022314X&md5=c4d3f2071d49b780c4dc90bed8722d3b" title="Click to view the MathML source">−π<α≤π$-\pi <\alpha \le \pi$.