F. T. Howard

Degenerate weighted Stirling numbers

Abstract We define the degenerate weighted Stirling numbers of the first and second kinds, S 1 ( n , k , λ ∥ θ ) and S ( n , k , λ ∥ θ ). By specializing λ and θ we can obtain the Stirling numbers, the weighted Stirling numbers and the degenerate Stirling numbers. Basic properties of S 1 ( n , k , λ ∥ θ ) and S ( n , k , λ ∥ θ ), such as recurrence formulas and combinatorial interpretations, are presented, and a theorem which relates S 1 ( n , k , λ ∥ θ ) and S ( n , k , λ ∥ θ ) to each other, and to other special numbers, is proved. This theorem provides a unified approach to a number of special cases which have recently appeared in the literature.

Explicit formulas for degenerate Bernoulli numbers

Abstract The ‘degenerate’ Bernoulli numbers β m ( λ ) can be defined by means of the exponential generating function x ((1 + λx ) 1/ λ −1) −1 . L. Carlitz proved an analogue of the Staudt-Clausen theorem for these numbers, and he showed that β m ( λ ) is a polynomial in λ of degree ⩽ m . In this paper we find explicit formulas for the coefficients of the polynomial β m ( λ ), and we give a new proof of the degenerate Staudt-Clausen theorem. New recursion formulas for β m ( λ ) are also proved.

A special class of Bell polynomials

and we prove similar equations involving the Bernoulli numbers, the van der Pol numbers, and the numbers generated by the reciprocal of ex x 1. (All of these special numbers are defined in Section 2.) Thus, the V(n, k) provide a link between these special numbers which is not obvious. In Section 5 we look, more generally, at the Bell polynomials 3fn,k(al, a2, 3 a, 4 a, 5 a, . . ); and we show how the results of Section 4 can

Polynomials related to the Bessel functions

In this paper we examine the polynomials Wn(a) defined by means of 4exa[x(ex 1) 2(eX + 1 = Wn(a)xnln!. n=0 These polynomials are closely related to the zeros of the Bessel function of the first kind of index -3/2, and they are in some ways analogous to the Bernoulli and Euler polynomials. This analogy is discussed, and the real and complex roots of Wn(a) are investigated. We show that if n is even then Wn(a) > 0 for all a, and if n is odd then Wn(a) has only the one real root a = 1/2. Also we find upper and lower bounds for all b such that Wn(a + bi) = 0. The problem of multiple roots is discussed and we show that if n 0, 1, 5, 8 or 9 (mod 12), then Wn(a) has no multiple roots. Finally, if n 0, 1, 2, 5, 6 or 8 (mod 12), then Wn(a) has no factor of the form a2 + ca + d where c and d are integers.

Factors and roots of the van der Pol polynomials

The van der Pol polynomials Vn(a) are defined by means of n x3exa[6x(ex + 1)12(eX 1)]1 = E V (a)xn/n!. n=O n In this paper new properties of these polynomials are derived. It is shown that neither V2n (a) nor V2n+1(a)/(a 1/2) has rational roots, and that if n = 2 * 3m, m > 0, or n = 3m + 3t, m > t > 0, or n m(p -3), p a prime number, 3m < p, then Vn(a) and Vn+1(a)/(a 1/2) are both irreducible over the rational field. It is also shown that if n = 2then V (a) is irreducible over the rational field. Finally, possible factors of the van der Pol polynomials are discussed.

A General Lacunary Recurrence Formula

The Bernoulli numbers B n may be defined by means of the generating function

Some sequences of rational numbers related to the exponential function

\frac{x}{{{e^{^x}} - 1}} = \sum\limits_{n = 0}^\infty {{B_n}} \frac{{{x^n}}}{{n!}}

A theorem relating potential and bell polynomials

(1.1) . An example of a “lacuary” recurrence for these numbers is