Log-concavity of stirling numbers and unimodality of stirling distributions

A series of inequalities involving Stirling numbers of the first and second kinds with adjacent indices are obtained. Some of them show log-concavity of Stirling numbers in three different directions. The inequalities are used to prove unimodality or strong unimodality of all the subfamilies of Stirling probability functions. Some additional applications are also presented.     

Non-central stirling numbers and some applications

Non-central Stirling numbers of the first and second kind are introduced and corresponding representations and recurrences are given along with some applications in occupancy problems and discrete distribution theory.     

Factorial functions and stirling numbers of fractional orders

The aim of this paper is to study the binomial coefficients ( _n ^x ), the factorial polynomials [x]_n and [x]ⁿ, the Stirling numbers of first and second kind, namely s(n,k) and S(n,k), in the case that n ∈ ? is replaced by real α ∈ ?. In the course of the paper, the Vandermonde convolution formula is presented in an infinite series frame, the binomial coefficient function ( _a ^x ), α ∈ ?, is sampled in terms of the binomial coefficients ( _k ^x ) for k ∈ ?_o, Bell numbers of fractional orders are introduced. Emphasis is placed on the fractional order Stirling numbers s(α,k) and S(α,k), first studied here. Some applications of the S(α,k) are given.     

Periodicities of partition functions and stirling numbers modulo p

If p(n, k) is the number of partitions of n into parts ≤k, then the sequence {p(k, k), p(k + 1, k),…} is periodic modulo a prime p. We find the minimum period Q = Q(k, p) of this sequence. More generally, we find the minimum period, modulo p, of {p(n; T)}_{n ≥ 0}, the number of partitions of n whose parts all lie in a fixed finite set T of positive integers. We find the minimum period, modulo p, of {S(k, k), S(k + 1, k),…}, where these are the Stirling numbers of the second kind. Some related congruences are proved. The methods involve the use of cyclotomic polynomials over Z_p[x].     

Prime and prime power divisibility of Catalan numbers

For any prime p, the sequence of Catalan numbers

$\text{a}{}_{\mathrm{n}}\text{=}\text{1}\text{n}\text{2n?2}\text{n?1}$

is divided by the a_n prime to p into blocks B_k(k > 0) of a_n divisible by p. The lengths and positions of the B_k are determined. Additional results are obtained on prime power divisibility of Catalan numbers.     

Calculating formula and divisibility for relative class numbers of abelian function fields

In this paper abelian function fields are restricted to the subfields of cyclotomic function fields. For any abelian function field K/k with conductor an irreducible polynomial over a finite field of odd characteristic, we give a calculating formula of the relative divisor class number of K. And using the given calculating formula we obtain a criterion for checking whether or not the relative divisor class number is divisible by the characteristic of k.     

Generalization of Kummer's criterion for divisibility of class numbers

We give a necessary and sufficient condition for the relative class number of an imaginary field contained in Q(e^2πi/p?) to be divisible by p. We also give a sufficient condition for the class number of a real field contained in Q(e^2πi/p?) not to be divisible by p.     

A probabilistically attained set of polynomials that generate stirling numbers of the second kind

Using probabilistic arguments, we derive a sequence of polynomials in one variable which generate the Stirling numbers of the second kind. Specifically, S^m_c=(c!/m!)P_c-m(c), where S^m_c is the desired Stirling number and P_c-m(·) is the polynomial of degree c-m.     

A property of algebraic univoque numbers

Consider the set $ {\mathcal{U}} $ of real numbers q ≧ 1 for which only one sequence (c _i ) of integers 0 ≦ c _i ≦ q satisfies the equality Σ _i=1 ^∞ c _i q ^?i = 1. We show that the set of algebraic numbers in $ {\mathcal{U}} $ is dense in the closure $ \overline {\mathcal{U}} $ of $ {\mathcal{U}} $ .     

On some 2-adic properties of a recurrence involving stirling numbers

We analyze some 2-adic properties of the sequence defined by the recurrence Z(1) = 1; Z(n) = Σ _k=1 ⁿ⁻¹ S(n, k)Z(k), n ≥ 2, which counts the number of ultradissimilarity relations, i.e., ultrametrics on an n-set. We prove the 2-adic growth property ν ₂(Z(n)) ≥ ⌈log₂ n⌉ −1 and present conjectures on the exact values.     

Decidability of the universal theory of natural numbers with addition and divisibility

The class of all quantifier-free formulas constructed from atomic formulas of the form (x+y= z),(x=1), and (x¦y) is considered, where the predicate symbol ¦ is interpreted as the divisibility relation on nonnegative integers. The decidability isproved of the set of all formulas of this form which are true for at least one choice of values for the variables. This result is equivalent to the decidability of the universal theory of natural numbers with addition and divisibility.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Mathematicheskogo Instituta im. V. A. Steklova Akad. Nauk SSSR, Vol. 60, pp. 15–28, 1976. Results presented September 26, 1974.The author wishes to thank his advisor N. K. Kosovskii, who suggested the topic of this paper, for his aid in checking and formulating these results.     

A property of Pisot numbers and Fourier transforms of self-similar measures

For any Pisot number β it is known that the set F (β)={t:lim n→∞‖tβ n‖= 0} is countable,where a is the distance between a real number a and the set of integers.In this paper it is proved that every member in this set is of the form cβ n,where ‖n‖ is a nonnegative integer and c is determined by a linear system of equations.Furthermore,for some self-similar measures μ associated with β,the limit at infinity of the Fourier transforms lim n→∞μ(tβ n)≠0 if and only if t is in a certain subset of F (β).This generalizes a similar result of Huang and Strichartz.     

On an approximation property of Pisot numbers

Let 1<q<2 be a real number, m≥1 be a rational integer and l_m(q)={|P(q)|,P∈Z[X],P(q)≠0,H(P)≤m}, where Z[X] denotes the set of polynomials P with rational integer coefficients and H(P) is the height of P. The value of l_m(q) was determined for many particular Pisot numbers ([3] and [7]). In this paper we determine the infimum and the supremum of the numbers l_m(q) for any fixed m. We also determine the greatest limit point for the case m=1. This revised version was published online in June 2006 with corrections to the Cover Date.