A generating function technique for Beatty sequences and other step sequences. (English) Zbl 1044.11009
The author presents a technique for expressing generating functions of sequences of integers defined using a real parameter \(x\), as for instance \(\sum_{n=1}^\infty z^{\lfloor n/x\rfloor}\), \(\sum_{n=1}^\infty \lfloor{n\over x}+1\rfloor\), etc.
The main theorem of the paper says: Let \(g(x,n)\), with \(x\in{\mathbb R}^+\), be a simple step complex valued function for each \(n\). Let \(\sum_{n=1}^\infty V_{(0,x)}(g(\cdot,n))<\infty\) for each \(x\), where \(V_I(g)\) denotes the variation of \(g\) on the interval \(I\). Let \(D\) be any set containing \(\{r; g(\cdot,n)\text{ is not continuous at }r\) for some \(n\)
The main theorem of the paper says: Let \(g(x,n)\), with \(x\in{\mathbb R}^+\), be a simple step complex valued function for each \(n\). Let \(\sum_{n=1}^\infty V_{(0,x)}(g(\cdot,n))<\infty\) for each \(x\), where \(V_I(g)\) denotes the variation of \(g\) on the interval \(I\). Let \(D\) be any set containing \(\{r; g(\cdot,n)\text{ is not continuous at }r\) for some \(n\)
Reviewer: Štefan Porubský (Praha)
MSC:
11B83 | Special sequences and polynomials |
05A15 | Exact enumeration problems, generating functions |
05A17 | Combinatorial aspects of partitions of integers |
11B34 | Representation functions |
11B75 | Other combinatorial number theory |
11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |
11M35 | Hurwitz and Lerch zeta functions |
Keywords:
Lambert series; Beatty sequence; complementary sequence; generating function; Fraenkel conjecture; Farey fractionOnline Encyclopedia of Integer Sequences:
Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < phi, where phi is the Golden ratio.Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < Pi.
Decimal expansion of Sum_(1/(2^q-1)) with the summation extending over all pairs of integers gcd(p,q) = 1, 0 < p/q < e = 2.718... .
Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>=1} A001951(n)/2^n.
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