Remarks on the Vietoris Sequence and Corresponding Convolution Formulas

Abstract

In this paper we consider the so-called Vietoris sequence, a sequence of rational numbers of the form $c_k=\frac{1}{2^k}\left( {\begin{array}{c}k\\ \lfloor \frac{k}{2}\rfloor \end{array}}\right)$, $k=0,1,\dots$ . This sequence plays an important role in many applications and has received a lot of attention over the years. In this work we present the main properties of the Vietoris sequence, having in mind its role in the context of hypercomplex function theory. Properties and patterns of the convolution triangles associated with ${(c_k)}_k$ are also presented.

Appendices

A Alternative Definitions

Vietoris sequence

Definition in terms of the generators of $\mathbb {H}$

The implementation of (3) requires the use of the free Mathematica package QuaternionAnalysis [13].

The function Quaternion defines a quaternion object, while QPower implements the usual quaternions powers. Both functions are included in the Mathematica package QuaternionAnalysis, where the arithmetic operations are also defined. The code of the function SymmetricPower is presented below. For more details on the use of the package we refer to the user guide included in the package documentation.

Double Factorial representation

Recursive definition

Pochhammer symbol representation

Alternating sum of a non-symmetric triangle

We point out that the function Ck[k,n] included in QuaternionAnalysis defines, for the choice $n=2$, the Vietoris sequence, using the form (5).

Gamma function representation

Integral representation

Catalan Numbers

Legendre Polynomials

Bessel functions

Bessel functions of the first kind with integer order are entire functions; here we have use the limit to avoid the indetermine form provided by a direct evaluation of the derivatives in Mathematica.

B Convolution Triangles

To produce the convolution triangle of a sequence in its rectangular form, one can use the function TriangleRect, in one of the following forms:

• 1. TriangleRect[{a0,a1,...,an},k]

  gives the $(n+1)\times (k+1)$ matrix corresponding to the first k convolutions of the sequence whose first $n+1$ terms are a0,...,an;

• 2. TriangleRect[exp,n,k]

  gives the $(n+1)\times (k+1)$ matrix corresponding to the first k convolutions of the sequence whose general term is given by the expression expr.

For example, the code

produces the matrix in the left hand side of Table 1. This result can also be obtained by using

The left justified form of the convolution triangle can be obtained by the use of the function TriangleLeft whose syntax is analogous to that of the function TriangleRect.

To obtain the table in the right hand side of Table 1 we just have to use:

or