Digit sums and generating functions

The Ramanujan Journal - We connect a primitive operation from arithmetic—summing the digits of a base-B integer—to q-series and product generating functions analogous to those in...     

Uniform partition extensions,a generating functions perspective

In this paper, a bivariate generating function CF(x, y) =f(x)-yf(xy)1-yis investigated, where f(x)= n 0fnxnis a generating function satisfying the functional equation f(x) = 1 + r j=1 m i=j-1aij xif(x)j.In particular, we study lattice paths in which their end points are on the line y = 1. Rooted lattice paths are defined. It is proved that the function CF(x, y) is a generating function defined on some rooted lattice paths with end point on y = 1. So, by a simple and unified method, from the view of lattice paths, we obtain two combinatorial interpretations of this bivariate function and derive two uniform partitions on these rooted lattice paths.     

The determination of generating functions and combinatorial sums by multidimensional residues

Siberian Mathematical Journal -     

Some generating functions of Jacobi polynomials

In this paper, Weisner’s group-theoretic method of obtaining generating functions is utilized in the study of Jacobi polynomialsP> _n ^(a,ß)(x) by giving suitable interpretations to the index (n) and the parameter (β) to find out the elements for constructing a six-dimensional Lie algebra.     

Some determinants of path generating functions

We evaluate four families of determinants of matrices, where the entries are sums or differences of generating functions for paths consisting of up-steps, down-steps and level steps. By specialisation, these determinant evaluations have numerous corollaries. In particular, they cover numerous determinant evaluations of combinatorial numbers—most notably of Catalan, ballot, and of Motzkin numbers—that appeared previously in the literature.     

Some general theorems on generating functions and their applications

Several general classes of generating functions are established for a certain sequence of functions defined by Equation (1) below. By suitably specializing the various parameters involved, each of these main results can be applied to yield known as well as new generating functions for such familiar orthogonal polynomials as Jacobi, Laguerre, Hermite, and Bessel polynomials, and also for numerous interesting generalizations of these polynomials studied in the literature.     

Some generating functions of modified Gegenbauer polynomials

L Weisner's group theoretic method has been introduced in the study of special function. In this paper we obtain two differential operators, one of which simultaneously raises the index and lowers the parameter of modified Gegenbauer polynomialsC _n ^v+n (x) by unity and the other acts onC _n ^v+n(x) in the reversed way by suitable interpretation to the indexn and the parameterv ofC _n ^v+n(x) . We have also found out the extended form of the groups generated by the operatorsA _ij(i,j=1,2). We have also derived some novel generating functions ofC _n ^v+n (x) from which several special generating functions can be easily derived.     

Some extensions of a property of linear representation functions

Let A={a₁,a₂,…}(a₁<a₂<) be an infinite sequence of nonnegative integers. Let k≥2 be a fixed integer and for , let R_k(A,n) be the number of solutions of a_i1++a_ik=n,a_i1,…,a_ikA, and let and denote the number of solutions with the additional restrictions a_i1<<a_ik, and a_i1≤≤a_ik respectively. Recently, Horváth proved that if d>0 is an integer, then there does not exist n₀ such that for n>n₀. In this paper, we obtain the analogous results for R_k(A,n), and .     

Some issues concerning approximations of functions by Fourier-Bessel sums

Some issues concerning the approximation of one-variable functions from the class $\mathbb{L}_2 $ by nth-order partial sums of Fourier-Bessel series are studied. Several theorems are proved that estimate the best approximation of a function characterized by the generalized modulus of continuity.     

Some remarks on a paper by L. Carlitz

We study a family of orthogonal polynomials which generalizes a sequence of polynomials considered by L. Carlitz. We show that they are a special case of the Sheffer polynomials and point out some interesting connections with certain Sobolev orthogonal polynomials.     

The algebra of generating functions for multiple divisor sums and applications to multiple zeta values

We study the algebra \({{\mathrm{{\mathcal {MD}}}}}\) of generating functions for multiple divisor sums and its connections to multiple zeta values. The generating functions for multiple divisor sums are formal power series in q with coefficients in \({\mathbb {Q}}\) arising from the calculation of the Fourier expansion of multiple Eisenstein series. We show that the algebra \({{\mathrm{{\mathcal {MD}}}}}\) is a filtered algebra equipped with a derivation and use this derivation to prove linear relations in \({{\mathrm{{\mathcal {MD}}}}}\). The (quasi-)modular forms for the full modular group \({{\mathrm{SL}}}_2({\mathbb {Z}})\) constitute a subalgebra of \({{\mathrm{{\mathcal {MD}}}}}\), and this also yields linear relations in \({{\mathrm{{\mathcal {MD}}}}}\). Generating functions of multiple divisor sums can be seen as a q-analogue of multiple zeta values. Studying a certain map from this algebra into the real numbers we will derive a new explanation for relations between multiple zeta values, including those of length 2, coming from modular forms.     

Some properties of generating functions obtained by Amann-Conley-Zehnder reduction

Consider an initial Lagrangian submanifold Λ₀ ⊂ T* ℝ ⁿ that admits a global generating function and a Hamiltonian isotopy Φ _H ^t . Then, we provide a global generating function for the Lagrangian submanifold Λ _t = Φ _H ^t (Λ₀) realized by applying the so-called Amann-Conley-Zehnder reduction. When Λ₀ is the zero-section, we study in some detail the asymptotic behavior of such generating functions and give an approximation result. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 33, Suzdal Conference-2004, Part 1, 2005.     

Some set partition statistics in non-crossing partitions and generating functions

In this paper we shall give the generating functions for the enumeration of non-crossing partitions according to some set partition statistics explicitly, which are based on whether a block is singleton or not and is inner or outer. Using weighted Motzkin paths, we find the continued fraction form of the generating functions. There are bijections between non-crossing partitions, Dyck paths and non-nesting partitions, hence we can find applications in the enumeration of Dyck paths and non-nesting partitions. We shall also study the integral representation of the enumerating polynomials for our statistics. As an application of integral representation, we shall give some remarks on the enumeration of inner singletons in non-crossing partitions, which is equivalent to one of udu's at high level in Dyck paths investigated in [Y. Sun, The statistic “number of udu's” in Dyck paths, Discrete Math. 284 (2004) 177-186].     

Some families of generating functions for the extended Srivastava polynomials

Almost four decades ago, H.M. Srivastava considered a general family of univariate polynomials, the Srivastava polynomials, and initiated a systematic investigation for this family [10]. In 2001, B. González, J. Matera and H.M. Srivastava extended the Srivastava polynomials by inserting one more parameter [4]. In this study we obtain a family of linear generating functions for these extended polynomials. Some illustrative results including Jacobi, Laguerre and Bessel polynomials are also presented. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials.     

Some contributions to the theory of cyclic quartic extensions of the rationals

K is a cyclic quartic extension of Q iff $\text{K = Q((rd + p d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext><mtext>)</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where d > 1, p and r are rational integers, d squarefree, for which p² + q² = r²d for some integer q. Following a paper of A. A. Albert we show that the absolute discriminant, $\text{d(}\text{K}\text{Q}\text{)}$, of the general cyclic quartic extension is given by $\text{d(}\text{K}\text{Q}\text{) = (W}{}^2\text{d}{}^2\text{)}$ for an explicitly computable rational integer W. We next find that the relative discriminant, $\text{d(}\text{K}\text{F}\text{)}$, is given by $\text{d(}\text{K}\text{F}\text{) = (W d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where $\text{F = Q (d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is K′s uniquely determined quadratic subfield. We use this last result in conjunction with Corollary 3, page 359, of Narkiewicz's “Elementary and Analytic Theory of Algebraic Numbers” (PWN-Polish Scientific Publishers, 1974) to establish the following Theorem 1: If the (wide) class number of$\text{F = Q(d}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$is odd then every cyclic quartic extensionKofQcontainingFhas a relative integral basis overF. We give a second, more organic, proof of Theorem 1 which also allows us to prove the following converse result, namely Theorem 2: Suppose the quadratic fieldFis contained in some cyclic quartic extension ofQand suppose thatFhas even (wide) class number. There then is a cyclic quartic extensionKofQcontainingFsuch thatKhas no relative integral basis overF.     

Some inverse theorems for approximation of functions by the Fourier-Laguerre sums

In this paper we prove two inverse theorems for approximation of functions of two variables by Fourier-Laguerre sums in the space L ₂(ℝ₊²;x ^α y ^β e ^−x−y ).