The VAN DER POL Numbers and a Related Sequence of Rational Numbers

Howard, F. T.

doi:10.1002/mana.19690420107

Cited by 12 publications

(7 citation statements)

References 4 publications

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“…Some results of this type for primes greater than 3 are found in [7]. Some results of this type for primes greater than 3 are found in [7].…”

Section: Introductionmentioning

confidence: 87%

Properties of the van der Pol numbers and polynomials.

1973

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…Some results of this type for primes greater than 3 are found in [7]. Some results of this type for primes greater than 3 are found in [7].…”

Section: Introductionmentioning

confidence: 87%

Properties of the van der Pol numbers and polynomials.

1973

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…for n \ 2 (see [8]), and also to the polynomials A k, n (z) associated with the numbers defined in (1.2): For n \ k,…”

Section: Recurrence Relations and Functional Equationsmentioning

confidence: 99%

“…An interesting relation among van der Pol numbers that has no analogue with the Bernoulli numbers is the following (see [8]): For r=1 this clearly reduces to (2.15), while for r=0 we obtain the wellknown fact that B 2n+1 =0 for n > 0. We note that (2.16) is not new; it was derived in [9], in a somewhat different notation.…”

Section: Now (211) Follows By Equating Coefficients Of T N Lmentioning

confidence: 99%

“…In the case of the ordinary Bernoulli numbers, the von Staudt-Clausen theorem tells us that p divides the denominator of B n if and only if p − 1 | n, so the first appearance is at n=p − 1. In the case of the van der Pol numbers V n = B (1,1) n , Howard [8] showed that the first appearance is at n=p − 3. This leads to the conjecture that in general, the first appearance of a prime p > r+s+1 > 1 in the denominator is at n=p − (r+s+1).…”

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confidence: 99%

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Arithmetic Properties of Bernoulli–Padé Numbers and Polynomials

Dilcher

Louise²

2002

Journal of Number Theory

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A class of generating functions based on the Padé approximants of the exponential function gives a doubly infinite class of number and polynomial sequences. These generalize the Bernoulli numbers and polynomials, as well as other sequences found in the literature. We derive analogues of the Kummer congruences, the von StaudtClausen Theorem, and other properties also satisfied by the ordinary Bernoulli numbers and polynomials.

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“…We conclude this section by defining the special numbers which are related to V(n, k). The van der Pol numbers have been the subject of several recent papers [8], [9].…”

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confidence: 99%

A Special Class of Bell Polynomials

Howard¹

1980

Mathematics of Computation

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Abstract. We examine the integers V(n, k) defined by means ofand, in particular, we show how these integers are related to the Bernoulli, Genocchi and van der Pol numbers, and the numbers generated by the reciprocal of e -x -1.We prove that the V(n, k) are also related to the numbers W(n, k) defined by fc! 2 W(n, k)x"/n< = [(* -2)(e* -l)]fc n=0 in much the same way the associated Stirling numbers are related to the Stirling numbers.Finally, we examine, more generally, the Bell polynomials Bn k(ax, a2, 3 -a, 4 -a, 5 -a, ... ) and show how the methods of this paper can be used to prove several formulas involving the Bernoulli and Stirling numbers.

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The VAN DER POL Numbers and a Related Sequence of Rational Numbers

Cited by 12 publications

References 4 publications

Properties of the van der Pol numbers and polynomials.

Properties of the van der Pol numbers and polynomials.

Arithmetic Properties of Bernoulli–Padé Numbers and Polynomials

A Special Class of Bell Polynomials

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