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We study an analogue of Riordan representation of Pascal matrices via Fibonomial coefficients. In particular,
we establish a relationship between the Riordan array and Fibonomial coefficients, and we show that such Pascal matrices can be represented
by an

Pascal matrices are infinite matrices whose entries are formed by binomial coefficients. Fibonomial coefficients are a certain class of generalized binomial coefficients, and its theory is now well understood. The Fibonomial coefficients are used to define Pascal matrices, called Pascal matrices via Fibonomial coefficients.

The Riordan group is quite easily developed but unifies many themes in enumeration. In the recent literature, special attention has been given to the concept of Riordan arrays, which is a generalization of the well-known Pascal triangle. Riordan arrays are infinite lower-triangular matrices defined by the generating function of their columns. They form a group, called the Riordan group (see [

Setting infinite dimensional Pascal matrix via Fibonomial coefficients requires very long applications and operations, whereas it was seen that these long applications and operations can be culminated more short and in a practical way through the aid of Riordan representation. Therefore, one can obtain Pascal matrix via Fibonomial coefficients through the aid of Riordan representation.

The aim of this study is to establish the Riordan representation of Pascal matrices via Fibonomial coefficients. It is confronted by a problem while obtaining Riordan representation. Using the usual operation can not be a solution for this problem. For overcoming this problem, a new binary operation is required to define. By using this operation,

The Fibonacci sequence is the starting point of our discussion. Thus, we briefly review some basic concepts and properties of Fibonomial coefficients. The Fibonacci numbers

The Riordan group is a set of infinite lower-triangular matrices each of which is defined by two generating functions, called a Riordan pair. Any infinite matrix of this group is called a Riordan array, and Riordan arrays are generalizations of Pascal's triangle. In fact, Pascal matrices via Fibonomial coefficients are Riordan arrays, and we will show that they can be represented by a Riordan pair. To this purpose we briefly review the Riordan group and we refer to [

Let

In this section, firstly

The generating functions of the Fibonomial coefficients [

Let

Let

The pair

(1) Closure. Indeed, for any two elements from

(2) Associativity is satisfied straightforwardly.

(3) An identity element is

Let

Therefore, we can write

The following theorem is analogous to the fundamental theorem of Riordan arrays.

Let

Let

Obtaining the entries of infinite dimensional Pascal matrix via Fibonomial coefficients requires cumbersome calculations. However, there is an alternative method using the Riordan group which appears to be more convenient. To this purpose, we start with the following theorem in which we obtain the

Let

We consider the infinite Pascal matrix via Fibonomial coefficients

Let

Let

Let

Let

The authors declare that there is no conflict of interests regarding the publication of this paper.