We study integer sequences and transforms that operate on them. Many of these transforms
are defined by triangular arrays of integers, with a particular focus on Riordan arrays and
Pascal-like arrays. In order to explore the structure of these transforms, use is made of
methods coming from the theory of continued fractions, hypergeometric functions, orthogonal
polynomials and most importantly from the Riordan groups of matrices. We apply the
Riordan array concept to the study of sequences related to graphs and codes. In particular,
we study sequences derived from the cyclic groups that provide an infinite family of colourings
of Pascal’s triangle. We also relate a particular family of Riordan arrays to the weight
distribution of MDS error-correcting codes. The Krawtchouk polynomials are shown to
give rise to many different families of Riordan arrays. We define and investigate Catalannumber-
based transformations of integer sequences, as well as transformations based on
Laguerre and related polynomials. We develop two new constructions of families of Pascallike
number triangles, based respectively on the ordinary Riordan group and the exponential
Riordan group, and we study the properties of sequences arising from these constructions,
most notably the central coefficients and the generalized Catalan numbers associated to
the triangles. New exponential-factorial constructions are developed to further extend this
theory. The study of orthogonal polynomials such as those of Chebyshev, Hermite, Laguerre
and Charlier are placed in the context of Riordan arrays, and new results are found. We
also extend results on the Stirling numbers of the first and second kind, using exponential
Riordan arrays. We study the integer Hankel transform of many families of integer sequences,
exploring links to related orthogonal polynomials and their coefficient arrays. Two particular
cases of power series inversion are studied extensively, leading to results concerning the
Narayana triangles.

A Study of Integer Sequences,Riordan Arrays, Pascal-like Arraysand Hankel Transforms

We study field diffeomorphisms $\Phi(x)= F(\rho(x))=a_0\rho(x)+a_1\rho^2(x)+\ldots=\sum_{j+0}^\infty a_j \rho^{j+1}$, for free and interacting quantum fields $\Phi$. We find that the theory is invariant under such diffeomorphisms if and only if kinematic renormalization schemes are used.

Diffeomorphisms of quantum fields

We recast homogeneous linear recurrence sequences with fixed coefficients in terms of partial Bell polynomials, and use their properties to obtain various combinatorial identities and multifold convolution formulas. Our approach relies on a basis of sequences that can be obtained as the INVERT transform of the coefficients of the given recurrence relation. For such a basis sequence $(y_n)$ with generating function $Y(t)$, and for any positive integer $r$, we give a formula for the convolved sequence generated by $Y(t)^r$ and prove that it satisfies an elegant recurrence relation.

Linear recurrence sequences and their convolutions via Bell polynomials

In this paper, we study degenerate complete and partial Bell polynomials and establish some new identities for those polynomials. In addition, we investigate the connections between modified degenerate complete and partial Bell polynomials, which are closely related to the degenerate complete and partial Bell polynomials, and the joint distribution of weighted sums of independent degenerate Poisson random variables.

/pdf/some-properties-of-degenerate-complete-and-partial-bell-1j9ez68cvf.pdf

Some properties of degenerate complete and partial Bell polynomials

The most frequently asked question in the $p-$adic lattice models of statistical mechanics is that whether a root of a polynomial equation belongs to domains $\mathbb{Z}_p^{*}, \ \mathbb{Z}_p\setminus\mathbb{Z}_p^{*}, \ \mathbb{Z}_p, \ \mathbb{Q}_p\setminus\mathbb{Z}_p^{*}, \ \mathbb{Q}_p\setminus\left(\mathbb{Z}_p\setminus\mathbb{Z}_p^{*}\right), \ \mathbb{Q}_p\setminus\mathbb{Z}_p, \ \mathbb{Q}_p $ or not. However, this question was open even for lower degree polynomial equations. In this paper, we give local descriptions of roots of cubic equations over the $p-$adic fields for $p>3$.

pdf/local-descriptions-of-roots-of-cubic-equations-over-p-adic-13q7sv9d88.pdf

Local Descriptions of Roots of Cubic Equations over P-adic Fields

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with nested compositions of partial Bell polynomials.

/pdf/some-convolution-identities-and-an-inverse-relation-3t2xd1ql8x.pdf

Some Convolution Identities and an Inverse Relation Involving Partial Bell Polynomials

of polynomials R[x] over R, namely the ring ^[[x]] of formal powers series in one variable over R, is hardly ever mentioned in such a course. In most cases, it is relegated to the homework problems (or to the exercises in the textbooks), and one learns that, like R[x], R[[x]] is an integral domain provided that R is an integral domain. More surprising is to learn that, in contrast to the situation of polynomials, in R[[x]] there are many invertible elements: while the only units in R[x] are the units of R, a necessary and sufficient condition for a power series to be invertible is that its constant term be invertible in R. This fact makes the study of arithmetic in /?[[*]] simple when R is a field: the only prime element is the variable x. As might be expected, the study of prime factorization in Z[[x]] is much more interesting (and complicated), but to the best of our knowledge it is not treated in detail in the available literature. After some basic considerations, it is apparent that the question of deciding whether or not an integral power series is prime is a difficult one, and it seems worthwhile to develop criteria to determine irreducibility in Z[[x]] similar to Eisenstein's criterion for polynomials. In this note we propose an easy argument that provides us with an infinite class of irreducible power series over Z. As in the case of Eisenstein's criterion in 7L\x\, our criteria give only sufficient conditions, and the question of whether or not a given power series is irreducible remains open in a vast array of cases, including quadratic polynomials. It is important to note that irreducibility in Z[x] and in Z[[x]] are, in general, un related. For instance, 6 + x + x2 is irreducible in Z[x] but can be factored in Z[[x]], while 2 + Ix + 3x2 is irreducible in Z[[x]] but equals (2 + x)(l +3x) as a polyno mial (observe that this is not a proper factorization in Z[[x]] since 1 + 3x is invert

Arithmetic in the Ring of Formal Power Series with Integer Coefficients

We prove an inverse relation and a family of convolution formulas involving partial Bell polynomials. Known and some presumably new combinatorial identities of convolution type are discussed. Our approach relies on an interesting multinomial formula for the binomial coefficients. The inverse relation is deduced from a parametrization of suitable identities that facilitate dealing with compositions of Bell polynomials.

Some convolution identities and an inverse relation involving partial Bell polynomials

pdf/colored-partitions-of-a-convex-polygon-by-noncrossing-4t7z08cxt0.pdf

Colored partitions of a convex polygon by noncrossing diagonals

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