A simple proof of Lerch's formula (Q2883390)

Let \(\alpha\) be a positive irrational number. In 1904 M. Lerch proposed as a question (No. 1547 in L'Intermédiaire des Mathématiciens) the following identity: NEWLINE\[NEWLINE\sum^{m}_{k=1} [k\alpha] + \sum^{[m\alpha]}_{k=1}[k \tfrac{1}{\alpha}] = m[m\alpha],NEWLINE\]NEWLINE where \([n]\) means the integer part of \(n\).NEWLINENEWLINEIn 1926 S. Beatty proposed as a problem (No. 3173 in Am. Math. Mon.) the following Lemma: Let \(x>1\) and \(y>1\) be irrational numbers such that \(\frac{1}{x}+\frac{1}{y}=1\). Then the sequences \(\left\{[nx]:n\geq1\right\}\) and \(\left\{[ny:\geq1]\right\}\) form a partition of the set of the natural numbers \(\left\{1, 2, 3, \ldots\right\}\).NEWLINENEWLINEThe authors of the present paper apply consequence of this lemma in their proof of the Lerch's formula.NEWLINENEWLINEAs a consequence of Lerch's formula the authors deduce the following identity: NEWLINE\[NEWLINE\sum^{F_{n}}_{k=1}([k\alpha]+k\alpha^{2})+ \sum^{F_{n+1}}_{k=1}[k\tfrac{1}{\alpha}]+\sum^{F_{n+2}}_{k=1}[k\tfrac{1}{\alpha^{2}}]= \sum^{F_{n}}_{k=1}[k\alpha^{3}]+ \sum^{F_{n+3}}_{k=1}[k\tfrac{1}{\alpha^{3}}]NEWLINE\]NEWLINE.