Riordan arrays and r-Stirling number identities

By using the theory of Riordan arrays, we establish four pairs of general r-Stirling number identities, which reduce to various identities on harmonic numbers, hyperharmonic numbers, the Stirling numbers of the first and second kind, the r-Stirling numbers of the first and second kind, and the r-Lah numbers. We further discuss briefly the connections between the r-Stirling numbers and the Cauchy numbers, the generalized hyperharmonic numbers, and the poly-Bernoulli polynomials. Many known identities are shown to be special cases of our results, and the combinatorial interpretations of several particular identities are also presented as supplements.     

Divisibility properties of hyperharmonic numbers

We extend Wolstenholme’s theorem to hyperharmonic numbers. Then, we obtain infinitely many congruence classes for hyperharmonic numbers using combinatorial methods. In particular, we show that the numerator of any hyperharmonic number in its reduced fractional form is odd. Then we give quantitative estimates for the number of pairs (n, r) lying in a rectangle where the corresponding hyperharmonic number \({ h_n^{(r)} }\) is divisible by a given prime number p. We also provide p-adic value lower bounds for certain hyperharmonic numbers. It is an open problem that given a prime number p, there are only finitely many harmonic numbers h _n which are divisible by p. We show that if we go to the higher levels r ≥  2, there are infinitely many hyperharmonic numbers \({ h_n^{(r)} }\) which are divisible by p. We also prove a finiteness result which is effective.     

On the maximum of r-Stirling numbers

Determining the location of the maximum of Stirling numbers is a well-developed area. In this paper we give the same results for the so-called r-Stirling numbers which are natural generalizations of Stirling numbers.     

Fuzzy n-cell numbers and the differential of fuzzy n-cell number value mappings

In this paper, we give the definition of a special kind of n-dimension fuzzy numbers, fuzzy n-cell numbers, discuss their operations and representation theorems, define a complete metric on the fuzzy n-cell number space and prove that the metric is equivalent to the supremum metric derived by the Hausdorff metric between the level sets of the n-dimension fuzzy numbers, and obtain an embedding theorem of the fuzzy n-cell number space (isometrically embeds it into a concrete Banach space). We also consider the differential of the fuzzy mappings from an interval into the fuzzy n-cell number space by using the embedding theorem.     

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.     

Structural properties of Dirichlet series with harmonic coefficients

An infinite family of functional equations in the complex plane is obtained for Dirichlet series involving harmonic numbers. Trigonometric series whose coefficients are linear forms with rational coefficients in hyperharmonic numbers up to any order are evaluated via Bernoulli polynomials, Gauss sums, and special values of L-functions subject to the parity obstruction. This in turn leads to new representations of Catalan’s constant, odd values of the Riemann zeta function, and polylogarithmic quantities. Consequently, a dichotomy result is deduced on the transcendentality of Catalan’s constant and a series with hyperharmonic terms. Moreover, making use of integrals of smooth functions, we establish Diophantine-type approximations of real numbers by values of an infinite family of Dirichlet series built from representations of harmonic numbers.     

Are the hyperharmonics integral? A partial answer via the small intervals containing primes

In a recent work, the authors have used Bertrand's postulate to give a partial answer to the conjecture of Mez? which says that the hyperharmonic numbers – iterations of partial sums of harmonic numbers – are not integers. In this Note, using small intervals containing prime numbers, we prove that a great class of hyperharmonic numbers are not integers.     

A generalization of the formula for the triangular number of the sum and product of natural numbers

This note generalizes the formula for the triangular number of the sum and product of two natural numbers to similar results for the triangular number of the sum and product of r natural numbers. The formula is applied to derive formula for the sum of an odd and an even number of consecutive triangular numbers.     

Some new Fibonacci and Lucas identities by matrix methods

The aim of this article is to characterize the 2 × 2 matrices X satisfying X ² = X + I and obtain some new identities concerning with Fibonacci and Lucas numbers.     

Applications of an Explicit Formula for the Generalized Euler Numbers

The authors establish an explicit formula for the generalized Euler NumbersE2n^（x）, and obtain some identities and congruences involving the higher＇order Euler numbers, Stirling numbers, the central factorial numbers and the values of the Riemann zeta-function.     

Prime and composite numbers as integer parts of powers

We study prime and composite numbers in the sequence of integer parts of powers of a fixed real number. We first prove a result which implies that there is a transcendental number ξ>1 for which the numbers [ξ^{n !}], n =2,3, ..., are all prime. Then, following an idea of Huxley who did it for cubics, we construct Pisot numbers of arbitrary degree such that all integer parts of their powers are composite. Finally, we give an example of an explicit transcendental number ζ (obtained as the limit of a certain recurrent sequence) for which the sequence [ζⁿ], n =1,2,..., has infinitely many elements in an arbitrary integer arithmetical progression. This revised version was published online in June 2006 with corrections to the Cover Date.     

Discrete fourier transform computation using prime Ramanujan numbers

Ramanujan numbers were introduced in [2] to implement discrete fourier transform (DFT) without using any multiplication operation. Ramanujan numbers are related to π and integers which are powers of 2. If the transform sizeN, is a Ramanujan number, then the computational complexity of the algorithms used for computing isO(N ²) addition and shift operations, and no multiplications. In these algorithms, the transform can be computed sequentially with a single adder inO(N ²) addition times. Parallel implementation of the algorithm can be executed inO(N) addition times, withO(N) number of adders. Some of these Ramanujan numbers of order-2 are related to the Biblical and Babylonian values of π [1]. In this paper, we analytically obtain upper bounds on the degree of approximation in the computation of DFT if JV is a prime Ramanujan number.     

On <Emphasis Type="Italic">l</Emphasis><Subscript>2,<Emphasis Type="Italic">p</Emphasis></Subscript>-circle numbers

Circle numbers are defined to reflect the Euclidean area-content and, for p ≠ 2, suitably defined non-Euclidean circumference properties of the l _2,p -circles, p ∈ [1, ∞]. The resulting function is continuous, increasing, and takes all values from [2, 4]. The actually chosen dual l _2,p -geometry for measuring the arc-length is closely connected with a generalization of the method of indivisibles of Cavalieri and Torricelli in the sense that integrating such arc-lengths means measuring area content. Moreover, this approach enables one to look in a new way into the co-area formula of measure theory which says that integrating Euclidean arc-lengths does not yield area content except for p = 2. The new circle numbers play a natural role, e.g., as norming constants in geometric measure representation formulae for p-generalized uniform probability distributions on l _2,p -circles.     

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _n=1^∞ F _2n−1⁻¹, ∑ _n=1^∞ F _2n−1⁻², ∑ _n=1^∞ F _2n−1⁻³ and write each ∑ _n=1^∞ F _2n−1^−s (s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.      

Symmetry p-adic invariant integral on ℤ p for Bernoulli and Euler polynomials

The main purpose of this paper is to investigate several further interesting properties of symmetry for the p-adic invariant integrals on ? _p . From these symmetry, we can derive many interesting recurrence identities for Bernoulli and Euler polynomials. Finally we introduce the new concept of symmetry of fermionic p-adic invariant integral on ? _p . By using this symmetry of fermionic p-adic invariant integral on ? _p , we will give some relations of symmetry between the power sum polynomials and Euler numbers. The relation between the q-Bernoulli polynomials and q-Dedekind type sums which discussed in Y. Simsek (q-Dedekind type sums related to q-zeta function and basic L-series, J. Math. Anal. Appl. 318 (2006), pp. 333–351) can be also derived by using the properties of symmetry of fermionic p-adic integral on ? _p .     

Carlitz’s q-Bernoulli and q-Euler numbers and polynomials and a class of generalized q-Hurwitz zeta functions

In this paper, we systematically recover the identities for the q-eta numbers η_k and the q-eta polynomials η_k(x), presented by Carlitz [L. Carlitz, q-Bernoulli numbers and polynomials, Duke Math. J. 15 (1948) 987–1000], which we define here via generating series rather than via the difference equations of Carlitz. Following a method developed by Kaneko et al. [M. Kaneko, N. Kurokawa, M. Wakayama, A variation of Euler’s approach to the Riemann zeta function, Kyushu J. Math. 57 (2003) 175–192] for a canonical q-extension of the Riemann zeta function, we investigate a similarly constructed q-extension of the Hurwitz zeta function. The details of this investigation disclose some interesting connections among q-eta polynomials, Carlitz’s q-Bernoulli polynomials -polynomials, and the q-Bernoulli polynomials that emerge from the q-extension of the Hurwitz zeta function discussed here.     

Hankel transforms of generalized Motzkin numbers

We study Hankel transform of the sequences (u,l,d),t, and the classical Motzkin numbers. Using the method based on orthogonal polynomials, we give closed‐form evaluations of the Hankel transform of the aforementioned sequences, sums of two consecutive, and shifted sequences. We also show that these sequences satisfy some interesting convolutional properties. Finally, we partially consider the Hankel transform evaluation of the sums of two consecutive shifted (u,l,d)‐Motzkin numbers. Copyright © 2017 John Wiley & Sons, Ltd.     

The Favoured Solutions of the Moment Problem

The sequence of Catalan numbers is known to be the moments of the Wigner law. A characterization of this sequence is given in terms of some determinants and the moment sequence of the q-Gaussian law is characterized in a similar fashion.     

New convolutions for complete and elementary symmetric functions

In this paper, we give new relationships between complete and elementary symmetric functions. These results can be used to discover and prove some identities involving r-Whitney numbers, Jacobi–Stirling numbers, Bernoulli numbers and other numbers that are specializations of complete and elementary symmetric functions.     

Several explicit formulae of Cauchy polynomials in terms of {r}-Stirling numbers

The integer values of Cauchy polynomials are expressed in terms of \({r}\)-Stirling numbers of the first kind. Several relations between the integral values of Bernoulli polynomials and those of Cauchy polynomials are obtained in terms of \({r}\)-Stirling numbers of both kinds. Also, we find a relation between the Cauchy polynomials and hyperharmonic numbers.