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In 1977 Foata proved bijectively, among other things, that the joint distribution of ascent and distinct nonzero value numbers on the set of subexcedant sequences is the same as that of descent and inverse descent numbers on the set of permutations, and the generating function of the corresponding bistatistics is the double Eulerian polynomial. In 2013 Foata’s result was rediscovered by Visontai as a conjecture, and then reproved by Aas in 2014. In this paper, we define a permutation code (that is, a bijection between permutations and subexcedant sequences) and show the more general result that two 5-tuples of set-valued statistics on the set of permutations and on the set of subexcedant sequences, respectively, are equidistributed. In particular, these results give another bijective proof of Foata’s result. [ABSTRACT FROM AUTHOR]

The aim of this paper is twofold. We first give a list of all Pisot polynomials of length at most 4. It contains seven polynomials of degree at most 5, and two infinite series of polynomials with unbounded degree. Then, for Pisot numbers α of length 3 and 4, we find explicitly the largest positive number L(α) such that for some ξ=ξ(α)∈R the limit points of the sequence of fractional parts {ξαn}n=1∞ all lie in the interval [L(α),1-L(α)]. [ABSTRACT FROM AUTHOR]

Abstract: The Cubic Sieve Method for solving the Discrete Logarithm Problem in prime fields requires a nontrivial solution to the Cubic Sieve Congruence (CSC) , where p is a given prime number. A nontrivial solution must also satisfy and , , where α is a given real number such that . The CSC problem is to find an efficient algorithm to obtain a nontrivial solution to CSC. CSC can be parametrized as and . In this paper, we give a deterministic polynomial-time ( bit-operations) algorithm to determine, for a given v, a nontrivial solution to CSC, if one exists. Previously it took time in the worst case to determine this. We relate the CSC problem to the gap problem of fractional part sequences, where we need to determine the non-negative integers N satisfying the fractional part inequality (θ and ϕ are given real numbers). The correspondence between the CSC problem and the gap problem is that determining the parameter z in the former problem corresponds to determining N in the latter problem. We also show in the case of CSC that for a certain class of primes the CSC problem can be solved deterministically in time compared to the previous best of . It is empirically observed that about one out of three primes is covered by the above class. [Copyright &y& Elsevier]

Abstract: We present a family of polynomials with nonnegative coefficients that are symmetric and have only real nonpositive zeros. Therefore, they are also log-concave, unimodal and have nonnegative γ-vectors. This family of polynomials arises by applying the operator to the rational function . Two distinguished examples are the classical Eulerian polynomials and the Narayana polynomials. We use two special linear bases, besides the standard basis, to provide various characterizations for these polynomials and their coefficients, in terms of generalized Stirling numbers of the second kind. The type of generalized Stirling numbers we deal with, emerge from the normal ordering of . We generalize several formulas known for Eulerian and Narayana polynomials. Likewise, we provide an explicit formula to compute the so-called γ-vector of any symmetric polynomial of the type consider in this paper. Further related polynomials are obtained by applying the operator to a general Möbius transformation. Along the way, associated families of novel nonnegative integer sequences are put in evidence as a byproduct of our polynomial characterizations. We provide explicit formulas for some of these integral sequences. [Copyright &y& Elsevier]

We introduce a new method to transform a knot diagram into a diagram of an unknot using a polynomial representation of the knot. Here the unknotting sequence of a knot diagram with least number of crossing changes can be realized by a family of polynomial maps. The number of singular knots in this family is defined to be the singularity index of the diagram. We show that the singularity index of a diagram is always less than or equal to its unknotting number. [ABSTRACT FROM AUTHOR]

This is the fifth and last in our series of notes concerning some classical inequalities such as the Ostrowski, Simpson, Iyengar, and Ostrowski–Grüss inequalities in . In the last note, we propose an improvement of the Ostrowski–Grüss inequality which involves knots where is an arbitrary numbers. More precisely, suppose that , and are arbitrary sequences with and . The main result of the present paper is to estimate in terms of either or . Unlike the standard Ostrowski–Grüss inequality and its known variants which basically estimate in terms of a correction term as a linear polynomial of x and some derivatives of f, our estimate allows us to freely replace and the correction term by using knots and . As far as we know, this is the first result involving the Ostrowski–Grüss inequality with three sequences of parameters. [ABSTRACT FROM AUTHOR]

A composition of a positive integer in which a part of size n may be assigned one of n colors is called an n-color composition. Let a denote the number of n-color compositions of the integer m. It is known that a= F for all m≥1, where F denotes the Fibonacci number defined by F= F+ F if m≥2, with F=0 and F=1. A statistic is studied on the set of n-color compositions of m, thus providing a polynomial generalization of the sequence F. The statistic may be described, equivalently, in terms of statistics on linear tilings and lattice paths. The restriction to the set of n-color compositions having a prescribed number of parts is considered and an explicit formula for the distribution is derived. We also provide q-generalizations of relations between a and the number of self-inverse n-compositions of 2 m+1 or 2 m. Finally, we consider a more general recurrence than that satisfied by the numbers a and note some particular cases. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to deal with applications of umbral calculus on fermionic p-adic integral on . From those applications, we derive some new identities on Genocchi numbers and polynomials. Moreover, a systemic study of the class of Sheffer sequences on the generating function of Genocchi polynomials are given. [Copyright &y& Elsevier]

This paper bounds the computational cost of computing the Kauffman bracket of a link in terms of the crossing number of that link. Specifically, it is shown that the image of a tangle with g boundary points and n crossings in the Kauffman bracket skein module is a linear combination of O(29) basis elements, with each coefficient a polynomial with at most n non-zero terms, each with integer coefficients, and that the link can be built one crossing at a time as a sequence of tangles with maximum number of boundary points bounded by C√n for some C. From this it follows that the computation of the Kauffman bracket of the link takes time and memory a polynomial in n times (2C√n. [ABSTRACT FROM AUTHOR]

The row first-minus-last right (RFMLR) circulant matrix and row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the exact formulae of determinants of the two pattern matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas sequences in terms of finite many terms of these sequences. [ABSTRACT FROM AUTHOR]

We study combinatorial and number-theoretic issues arising from group-invariant CR mappings from spheres to hyperquadrics. Given an initial complex analytic function, we define two sequences of complex numbers. We derive formulae for these sequences in terms of the Taylor coefficients of the initial function and, in the polynomial case, also in terms of the reciprocals of the roots. We connect these formulae to CR mappings invariant under arbitrary representations of finite cyclic subgroups of the unitary group. We prove that the resultant of them-th polynomial (which in all cases has integer coefficients) and any of its partial derivatives is divisible bymm. [ABSTRACT FROM AUTHOR]

Abstract: This article deals with the introduction of Appell type sets of polynomials and associated numbers. The 2-iterated Appell polynomials are introduced and the Bernoulli and Euler based Appell polynomials are deduced as their particular cases. Different sets of polynomials, namely 2-iterated Bernoulli and Euler, Bernoulli–Euler (or Euler–Bernoulli) and their related numbers are considered. The operational relations between these families and Appell polynomials are used on the results of the Bernoulli and Euler polynomials to obtain the results for the corresponding mixed polynomials. [Copyright &y& Elsevier]

To count the number of chemical compositions of a particular mass, we consider an alphabet $${\mathop{\bf a}}$$ with a mass function which assigns a mass to each letter in $${\mathop{\bf a}}$$ . We then compute the mass of a word (an ordered sequence of letters) by adding the masses of the constituent letters. Our main interest is to count the number of words that have a particular mass, where we ignore the order of the letters within the word. We show first that counting the number of words of a given mass has a geometric interpretation, whose solutions are called Ehrhart quasi-polynomials, a class of functions defined on integers. These special functions are 'periodic' in the sense that they use the same polynomial every λ steps. In addition to discovering the connection between counting compositions and Ehrhart quasi-polynomials, we also find number theoretic results that greatly reduce the number of candidates for the period, λ. Finally, we illustrate the usefulness of these results and the use of a software library named barvinok (by Verdoolaege et al.) by applying them to eight different classes of chemical compositions, including organic molecules, peptides, DNA, and RNA. [ABSTRACT FROM AUTHOR]

Abstract: A sequence of polynomials in , such that has positive leading coefficient, each is irreducible in and for all i, is defined to be a polynomial Cunningham chain of the first or second kind, depending on whether or −1 respectively. If k is the least positive integer such that is reducible in , then we say the chain has length k. In the previous article, “Polynomial Cunningham chains”, a proof is given that there exist infinitely many polynomial Cunningham chains of length k of either kind. It is also deduced from this result that there exist infinitely polynomial Cunningham chains of infinite length of either kind. However, the proof contains an error. In this article we give correct proofs of these statements. [Copyright &y& Elsevier]

Let f(n) denote the number of relatively prime sets in {1, … , n}. This is sequence A085945 in Sloane's On-Line Encyclopedia of Integer Sequences. Motivated by a paper of Ayad and Kihel [1], we show that there are at most finitely many positive integers n such that f(n) is a perfect power of exponent > 1 of some other integer. We also show that the sequence {f(n)} n≥1 is not holonomic; that is, it satisfies no recurrence relation of finite order with polynomial coefficients. [ABSTRACT FROM AUTHOR]

Let Bn(t) be an nth Stern polynomial and let e(n) = deg Bn(t) be its degree. In this note we continue our study started in [On certain arithmetic properties of Stern polynonials, Publ. Math. Debrecen79(1-2) (2011) 55-81] of the arithmetic properties of the sequence of Stern polynomials and the sequence $\{e(n)\}_{n = 1}^{\infty}$. We also study the sequence d(n) = ordt=0 Bn(t). Among other things we prove that d(n) = ν(n), where ν(n) is the maximal power of 2 which divides the number n. We also count the number of the solutions of the equations e(m) = i and e(m) - d(m) = i in the interval [1, 2n]. We also obtain an interesting closed expression for a certain sum involving Stern polynomials. [ABSTRACT FROM AUTHOR]

Let f( x) be the product of several linear polynomials with integer coefficients. In this paper, we obtain the estimate log lcm ( f(1),..., f( n))∼ An as n→∞, where A is a constant depending on f. [ABSTRACT FROM AUTHOR]

In this same journal, Coons published recently a paper [The transcendence of series related to Stern's diatomic sequence, Int. J. Number Theory6 (2010) 211-217] on the function theoretical transcendence of the generating function of the Stern sequence, and the transcendence over ℚ of the function values at all non-zero algebraic points of the unit disk. The main aim of our paper is to prove the algebraic independence over ℚ of the values of this function and all its derivatives at the same points. The basic analytic ingredient of the proof is the hypertranscendence of the function to be shown before. Another main result concerns the generating function of the Stern polynomials. Whereas the function theoretical transcendence of this function of two variables was already shown by Coons, we prove that, for every pair of non-zero algebraic points in the unit disk, the function value either vanishes or is transcendental. [ABSTRACT FROM AUTHOR]

Abstract: We present applications of matrix methods to the analytic theory of polynomials. We first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials. Then we use matrix methods to establish a new log-majorization result on roots of polynomials. The theory of multiplier sequences gives the common link between the applications. [Copyright &y& Elsevier]

Abstract: A set is -recognizable for an abstract numeration system , if the set of its representations is accepted by a finite automaton. We show that the growth function of an -recognizable set is always either where and , or , where with . If the number of words of length in the numeration language is bounded by a polynomial, then the growth function of an -recognizable set is , where with . Furthermore, for every with , we can provide an abstract numeration system built on a polynomial language and an -recognizable set such that the growth function of is . For all positive integers and , we can also provide an abstract numeration system built on an exponential language and an -recognizable set such that the growth function of is . [Copyright &y& Elsevier]

Abstract: The objective of this paper is the study of functions which only act on the digits of an expansion. In particular, we are interested in the asymptotic distribution of the values of these functions. The presented result is an extension and generalization of a result of Bassily and Kátai to number systems defined in a quotient ring of the ring of polynomials over the integers. [Copyright &y& Elsevier]

Abstract: The well-known Faà di Bruno formula for higher derivatives of a composite function plays an important role in combinatorics. In this paper we obtain a unified approach to some recurrence sequences for the complete Bell polynomials by using Faà di Bruno’s formula. Furthermore, we obtain an inverse relation of Lah numbers of two types, and by using this relation we obtain some new and interesting combinatorial identities. [Copyright &y& Elsevier]

Two methods to test whether a given polynomial has two distinct roots whose quotient is a root of unity are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the relationship between the generation polynomial of a linear sequence and its decimation sequence over the finite field 픽2. Then we propose an upper bound of the number of terms in the generation polynomial of a decimation sequence of a linear sequence whose generation polynomial is trinomial. Finally, we suggest a method to calculate the terms of a decimation polynomial and their number directly, which can be used in the construction of irreducible polynomials with a controlled number of terms, when the source trinomial and the decimation distance satisfy a certain condition. [ABSTRACT FROM AUTHOR]

Abstract: Duistermaat and van der Kallen show that there is no nontrivial complex Laurent polynomial all of whose powers have a zero constant term. Inspired by this, Sturmfels poses two questions: Do the constant terms of a generic Laurent polynomial form a regular sequence? If so, then what is the degree of the associated zero-dimensional ideal? In this note, we prove that the Eulerian numbers provide the answer to the second question. The proof involves reinterpreting the problem in terms of toric geometry. [ABSTRACT FROM AUTHOR]

Abstract: Given a set of primes P, we determine the necessary and sufficient criterions for the existence of a sequence of polynomials Γ, with support base P, which is a solution of the functional equations arising from multiplication of quantum integers discussed in Melvyn B. Nathanson (2003) and which cannot be generated by quantum integers. [Copyright &y& Elsevier]

Abstract: From a delta series and its compositional inverse , Hsu defined the generalized Stirling number pair . In this paper, we further define from and the generalized higher order Bernoulli number pair . Making use of the Bell polynomials, the potential polynomials as well as the Lagrange inversion formula, we give some explicit expressions and recurrences of the generalized higher order Bernoulli numbers, present the relations between the generalized higher order Bernoulli numbers of both kinds and the corresponding generalized Stirling numbers of both kinds, and study the relations between any two generalized higher order Bernoulli numbers. Moreover, we apply the general results to some special number pairs and obtain series of combinatorial identities. It can be found that the introduction of generalized Bernoulli number pair and generalized Stirling number pair provides a unified approach to lots of sequences in mathematics, and as a consequence, many known results are special cases of ours. [Copyright &y& Elsevier]

Abstract: Let D be the ring of integers of a number field K, P a prime of D for which is finite, the corresponding valuation of K and E a homogeneous subset of D with respect to P, i.e. a set with the property for some positive integer ℓ. Also let denote the ring of polynomials in which take values in D when evaluated at points of E. The characteristic sequence of E with respect to P is the sequence of integers where is the fractional ideal formed by 0 and the leading coefficients of elements of of degree ⩽n. In this paper we give a recursive method for computing the limit for any homogeneous set, apply it to the special case of the homogeneous sets for  , and show that in general the possible values of this limit as E ranges over all possible homogeneous subsets are dense in the interval . We also apply this method to certain infinite unions of homogeneous sets and obtain formulas for these limits as regular continued fractions. [Copyright &y& Elsevier]

Abstract: We examine a recolouring scheme ostensibly used to assist in classifying geographic data. Given a drawing of a graph with bi-chromatic points, where the points are the vertices of the graph, a point can be recoloured if it is surrounded by neighbours of the opposite colour. The notion of surrounded is defined as a contiguous subset of neighbours that span an angle greater than 180 degrees. The recolouring of surrounded points continues in sequence, in no particular order, until no point remains surrounded. We show that for some classes of graphs the process terminates in a polynomial number of steps. On the other hand, there are classes of graphs with infinite recolouring sequences. [Copyright &y& Elsevier]

We completely classify all polynomials of type (x2 + x -(Δ-1))/4 which are prime or 1 for a range of consecutive integers x ≥ 0, called Rabinowitsch polynomials, where Δ ≡ 1(mod4) with Δ > 1 square-free. This corrects, extends, and completes the results by Byeon and Stark (2002, 2003) via the use of an updated version of what Andrew Granville has dubbed the Rabinowitsch-Mollin-Williams Theorem—by Granville and Mollin (2000) and Mollin (1996). Furthermore, we verify conjectures of this author and pose more based on the new data. [ABSTRACT FROM AUTHOR]

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a generalmethod to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented. [ABSTRACT FROM AUTHOR]

Abstract: We associate in a canonical way a substitution to any abstract numeration system built on a regular language. In relationship with the growth order of the letters, we define the notion of two independent substitutions. Our main result is the following. If a sequence x is generated by two independent substitutions, at least one being of exponential growth, then the factors of x appearing infinitely often in x appear with bounded gaps. As an application, we derive an analogue of Cobham''s theorem for two independent substitutions (or abstract numeration systems) one with polynomial growth, the other being exponential. [Copyright &y& Elsevier]

Let q = pm where p is an odd prime, m ≥ 2, and 1 ≤ k ≤ m - 1. Let Tr be the trace mapping from Fq to Fp and (ζp = e 2π i/p be a primitive pth root of unity. In this paper, we determine the value distribution of the following exponential sums: Σ x ϵ Fq χ (αxpk+1 + βx²) (α, β ϵ Fq) where χ(x) = ζpTr(x) is the canonical additive character of Fq. As applications, we have the following. 1) We determine the weight distribution of the cyclic codes C1 and C2 over Fpt with parity-check polynomial h2 (x)h3 (x) and h1 (x)h2 (x)h3 (x), respectively, where t is a divisor of d = gcd (m, k), and h1 (x), h2 (x), and h3 (x) are the minimal polynomials of π-1, π-2, and π-(Pk+1) over FPt respectively, for a primitive element π of Fq. 2) We determine the correlation distribution between two m-sequences of period q -1. Moreover, we find a new class of p-ary bent functions. This paper extends the results in Feng and Luo (2008). [ABSTRACT FROM AUTHOR]

There are two analytic approaches to Bernoulli polynomials Bn(x): either by way of the generating function zexz/(ez - 1) = ∑ Bn(x)zn/n! or as an Appell sequence with zero mean. In this article, we discuss a generalization of Bernoulli polynomials defined by the generating function zN exz/(ez - TN-1(z)), where TN(z) denotes the Nth Maclaurin polynomial of ez, and establish an equivalent definition in terms of Appell sequences with zero moments in complete analogy to their classical counterpart. The zero-moment condition is further shown to generalize to Bernoulli polynomials generated by the confluent hypergeometric series. [ABSTRACT FROM AUTHOR]

The q-Catalan numbers studied by Carlitz and Riordan are polynomials in q with nonnegative coefficients. They evaluate, at q = 1, to the Catalan numbers: 1, 1, 2, 5, 14,..., a log-convex sequence. We use a combinatorial interpretation of these polynomials to prove a q-log-convexity result. The sequence of q-Catalan numbers is not q-log-convex in the narrow sense used by other authors, so our work suggests a more flexible definition of q-log convex be adopted. [ABSTRACT FROM AUTHOR]

Let β be a fixed element of 픽q((X-1)) with polynomial part of degree ≥ 1, then any formal power series can be represented in base β, using the transformation Tβ : f ↦ {βf} of the unit disk $\mathcal{M}_0$. Any formal power series in $\mathcal{M}_0$ is expanded in this way into dβ(f) = (ai(X))i≥1, where $a_i(X)=[\beta T^{i-1}_\beta(f)]\in \mathbb{F}_q[X]$. The main aim of this paper is to characterize the formal power series β(|β| > 1), such that dβ(1) is finite, eventually periodic or automatic (such characterizations do not exist in the real case). [ABSTRACT FROM AUTHOR]

We describe a method for establishing that values from a fixed polynomial sequence are represented frequently by some prescribed sum of powers of natural numbers. As an illustration of this method, we show that for at least X129/136 of the integers n with 1 ≤ n ≤ X, a fixed quadratic polynomial φ(n) may be written as the sum of five cubes of positive integers. A similar result is established for the sum of a square and three cubes of positive integers. [ABSTRACT FROM PUBLISHER]

Let f(x) be an integer-valued polynomial with no fixed integer divisor ≥ 2, that is, for no integer d ≥ 2 does d∣f(x) for all integers x. One generalization of the famous Waring problem is to determine whether, for large enough s, the equationf(x1)+f(x2)+…+f(xs)=n (1.1)is solvable in positive integers x1, …, xs for sufficiently large integers n. The existence of such s for every f was established by Kamke [5] in 1921. Subsequent authors (Pillai, Hua [2–4], Vinogradov, Načaev [7], and others) have studied the problem of bounding G(f), the least s for which (1.1) is solvable for all large n.Questions of local solubility of (1.1), that is, solubility of the congruencef(x1)+f(x2)+…+f(xs)≡n (mod q) (1.2)play a more important and complicated role in this problem than in the classical Waring problem. Let Γ0(f) denote the least number s so that (1.2) is solvable for every pair n, q. It is well known that Γ0(xk) ≤ 4k for every k, but Hua [4] found that for every k, the polynomial fk (x) = ∑ kj=1 (−1)k−j2j−1 (xj) satisfies Γ0(fk) ≥ 2k − 1 (take s = 2k−2, q = 2k and n = (−1)k in (1.2)). Clearly G(f) ≥ Γ0(f), but one can say more by restricting the values of n under consideration, as has been done by several authors in the case f(x) = x4 (for example [1, 6]).The singular series Ss,f(n) = ∑ ∞q=1 ∑qa=1(a,q)=1 (1/q ∑qr=1 e(af(r)/q))s e(−an/q), where e(z) = e2πiz, encapsulates the local solubility information. In particular, Ss, f(n) ≥ 0 for every n and Ss, f(n) > 0 if and only if (1.2) is soluble for every q. Define G(f) to be the least number s so that for every δ >0 and every n>n0(δ) with Ss, f(n) ≥ δ, (1.1) is soluble. The reason for taking Ss, f(n) ≥ δ instead of Ss, f(n) > 0 is that we wish to exclude from consideration certain n lying in sparse sequences for which (1.1) is insoluble but Ss, f(n) > 0. For example, taking f(x) = x4, s = 15 and nj = 79·16j(j = 0, 1, …), it can be shown that (1.1) is not soluble for n = nj, that Ss, f(nj) > 0 for all j, and that Ss, f(nj) → 0 as j → ∞. It is known that G(x4) = 16 (see [1]) and that G(x4) ≥ 11 almost holds (see [6]). [ABSTRACT FROM AUTHOR]

Using a work of Diaz concerning algebraic independence of certain sequences of numbers, we prove that if K ⊆ ℝ is a field of finite transcendence degree over the rationals, then every weakly o-minimal expansion of (K,≤,+,·) is polynomially bounded. In the special case where K is the field of all real algebraic numbers, we give a proof which makes use of a much weaker result from transcendental number theory, namely, the Gelfond–Schneider theorem. Apart from this we make a couple of observations concerning weakly o-minimal expansions of arbitrary ordered fields of finite transcendence degree over the rationals. The strongest result we prove says that if K is a field of finite transcendence degree over the rationals, then all weakly o-minimal non-valuational expansions of (K,≤,+,·) are power bounded. [ABSTRACT FROM AUTHOR]