The 𝐹-Analogue of Riordan Representation of Pascal Matrices via Fibonomial Coefficients

Abstract

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 𝐹-Riordan pair.

1. Introduction

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 [1]). Some of the main results on the Riordan group and its application to combinatorial sums and identities can be found in Sprugnoli (see [2, 3]).

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, 𝐹-analogue of Riordan representation is obtained. In particular, we show that Pascal matrices via Fibonomial coefficients of the first and the second kinds can be represented with an 𝐹-analogue of Riordan pair.

2. Preliminaries

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 𝐹_𝑛 are defined by the initial conditions 𝐹₀=0, 𝐹₁=1 and the recurrence 𝐹_𝑛=𝐹_𝑛−1+𝐹_𝑛−2 (1) for 𝑛≥2. Let 𝑛 and 𝑘 be integers with 𝑛≥𝑘≥0. Then, the Fibonomial coefficients are defined by 𝑛𝑘_𝐹=[𝑛]_𝐹![𝑘]_𝐹![𝑛−𝑘]_𝐹!, (2) where [𝑛]_𝐹!=𝐹_𝑛𝐹_𝑛−1⋅⋅⋅𝐹₁ and [0]_𝐹!=1. It can be shown that Fibonomial coefficients satisfy the following recursion relation: 𝑛𝑘_𝐹=𝐹_𝑘+1𝑘𝑛−1_𝐹+𝐹_{𝑛−𝑘−1}𝑛−1𝑘−1_𝐹 (3) (see [4–6]). Let 0≤𝑖,𝑗≤𝑛−1. The 𝑛×𝑛 Pascal matrix, 𝑃_𝑛=(𝑝_𝑖𝑗), is defined by 𝑝_𝑖𝑗=𝑖𝑗if0𝑖≥𝑗,otherwise. (4) The Pascal matrix via Fibonomial coefficients is denoted by 𝒫(𝑘,𝑚)=(𝑝_𝑖𝑗) and is defined by 𝑝_𝑖𝑗=𝑘+𝑗𝑚−1+𝑖−𝑗𝑘+𝑗𝑚−1_𝐹if0𝑖≥𝑗,otherwise, (5) where 𝑘, 𝑚∈ℕ and 𝑘+𝑗𝑚−1+𝑖−𝑗𝑘+𝑗𝑚−1𝐹 is the Fibonomial coefficient. Specially, 𝒫(1,1)=𝒫=(𝑝_𝑖𝑗) is defined by 𝑝_𝑖𝑗=𝑖𝑗_𝐹if0𝑖≥𝑗,otherwise. (6) Similarly, the generalized Pascal matrix via Fibonomial coefficients of the first kind, 𝑈_𝑛[𝑥]=(𝑈_𝑛(𝑥,𝑖,𝑗)), is defined by 𝑈_𝑛=𝑥𝑥,𝑖,𝑗^𝑖−𝑗𝑖𝑗_𝐹if0𝑖≥𝑗,otherwise, (7) and the generalized Pascal matrix via Fibonomial coefficients of the second kind, 𝑀_𝑛[𝑥]=(𝑀_𝑛(𝑥,𝑖,𝑗)), is defined by 𝑀_𝑛=𝑥𝑥,𝑖,𝑗^{𝑖+𝑗−2}𝑖𝑗_𝐹if0𝑖≥𝑗,otherwise. (8) Moreover, the extended generalized Pascal matrix via Fibonomial coefficients, Φ_𝑛[𝑥,𝑦]_𝐹=(𝜑_𝑛(𝑥,𝑦;𝑖,𝑗)_𝐹), is defined by 𝜑_𝑛𝑥,𝑦;𝑖,𝑗_𝐹=𝑥^𝑖−𝑗𝑦^{𝑖+𝑗−2}𝑖𝑗_𝐹if0𝑖≥𝑗,otherwise. (9) See [7–11] for details. Note that all of 𝒫, 𝑈_𝑛, 𝑀_𝑛, and Φ_𝑛 are 𝑛×𝑛 matrices.

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 [1–3, 12, 13] for a detailed treatment of the subject.

Definition 1 (see [1]). Let 𝑔 and 𝑓 be two functions defined by 𝑔(𝑥)=𝑔₀+𝑔₁𝑥+𝑔₂𝑥²𝑓(𝑥)+⋯,=𝑓₁𝑥+𝑓₂𝑥²+𝑓₃𝑥³+⋯ (10) with 𝑔₀≠0. Let us denote by (𝑔,𝑓) the infinite lower-triangular matrix whose 𝑗th column is formed by the coefficients of the power series 𝑔(𝑥)𝑓(𝑥)^𝑗,𝑗=0,1,2,…. (11) The first column of this matrix is called the 0th column. Let ℛ be the set of all infinite lower-triangular matrices defined by (11), and let (𝑔,𝑓) and (𝑢,𝑣)∈ℛ. Then ℛ becomes a group under the operation 𝑔,𝑓∗(𝑢,𝑣)∶=𝑔𝑢∘𝑓,𝑣∘𝑓. (12) In particular, ℛ is called the Riordan group and any element (𝑔,𝑓) of ℛ is called a Riordan pair. The identity element of ℛ is 𝐼=(1,𝑥). (13) And the inverse of any (𝑔,𝑓) is 𝑔,𝑓⁻¹=1𝑔∘𝑓,𝑓. (14) Here, 𝑓 is the compositional inverse of 𝑓; that is, 𝑓(𝑓(𝑥))=𝑓(𝑓(𝑥))=𝑥.

3. 𝐹-Analogue of FTRA

In this section, firstly ∗_𝐹 operation is defined and then using this operation a new theorem which is called 𝐹-analogue of FTRA is obtained.

The generating functions of the Fibonomial coefficients [14–18] are ^𝑛_𝑖=11−𝛼^{(𝑛−𝑖)}𝛽^(𝑖−1)𝑥=_𝑘≥0(−1)^{𝑘(𝑘+1)/2}𝑛𝑘_𝐹𝑥^𝑘,_𝑛𝑖=111−𝛼^(𝑖−1)𝛽^{(𝑛−𝑖)}𝑥=_𝑘≥0𝑘𝑛+𝑘−1_𝐹𝑥^𝑘, (15) where 𝛼=(1+√5)/2 and 𝛽=(1−√5)/2. For simplicity of notation, we denote these two generating functions as follows.

Definition 2. Let 𝑛≥0; then (1−𝑥):=^𝑛_𝐹^def𝑛_𝑖=11−𝛼^{(𝑛−𝑖)}𝛽^(𝑖−1)𝑥,1(1−𝑥)^𝑛_𝐹^def:=^𝑛_𝑖=111−𝛼^𝑖−1𝛽^𝑛−𝑖𝑥. (16)

Definition 3. Let ℱ denote the set of elements 𝑡𝑥^𝑚(1−𝑥)^𝑛_𝐹^def=𝑡𝑥^𝑚𝑛_𝑖=111−𝛼^𝑖−1𝛽^𝑛−𝑖𝑥=𝑡_𝑘≥0𝑛+𝑘−𝑚−1𝑛−1_𝐹𝑥^𝑘 (17) for all integers 𝑚,𝑛≥0 and 𝑡∈ℝ. Let ∗_𝐹∶ℱ×ℱ→ℱ be a binary operation defined as follows: 𝑡𝑥^𝑎(1−𝑥)^𝐴_𝐹∗_𝐹𝑢𝑥^𝑏(1−𝑥)^𝐵_𝐹^def=𝑡𝑥^𝑎1−𝛼^𝐵𝑥^𝐴_𝐹⋅𝑢𝑥^𝑏1−𝛽^𝐴𝑥^𝐵_𝐹=𝑡𝑢𝑥^𝑎+𝑏(1−𝑥)_𝐹^𝐴+𝐵. (18)

Lemma 4. The pair (ℱ,∗_𝐹) is a monoid.

Proof. (1) Closure. Indeed, for any two elements from ℱ we obtain an element from ℱ. That is, 𝑡𝑥^𝑎(1−𝑥)^𝐴_𝐹∗_𝐹𝑢𝑥^𝑏(1−𝑥)^𝐵_𝐹=𝑡𝑢𝑥^𝑎+𝑏(1−𝑥)_𝐹^𝐴+𝐵∈ℱ. (19)
(2) Associativity is satisfied straightforwardly.
(3) An identity element is 𝑡⁰𝑥⁰(1−𝑥)⁰_𝐹. (20)

Let 𝑔 and 𝑓 belong to ℱ with 𝑔(0)≠0 and 𝑓(0)=0. The infinite lower-triangular matrix whose 𝑗th column is formed by the coefficients of the power series is 𝑔(𝑥)∗_𝐹𝑓(𝑥)^[𝑗],𝑗=0,1,2,…, (21) where 𝑓(𝑥)^[𝑗]∶=𝑓(𝑥)∗_𝐹𝑓(𝑥)∗_𝐹⋯∗_𝐹𝑓(𝑥)_𝑗times. By using new ∗_𝐹 binary operation and (21), we obtain a representation which is the analogue of the Riordan representation. We call the representation 𝐹-analogue of Riordan representation and denote it by (𝑔,𝑓)_𝐹.

Therefore, we can write 𝑡(1−𝑥)^𝑛_𝐹𝐹∗_𝐹𝑢𝑥(1−𝑥)^𝑚_𝐹^[𝑗]=𝑡𝑢^𝑗𝑥^𝑗(1−𝑥)_𝐹^{𝑛+𝑚𝑗} (22) for any 𝑗≥0.

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

Theorem 5 (𝐹-analogue of FTRA). Let 𝑔(𝑥),𝑓(𝑥)∈ℱ with 𝑔(0)≠0 and 𝑓(0)=0. The 𝐹-analogue of the fundamental theorem of Riordan arrays is 𝑔(𝑥),𝑓(𝑥)_𝐹𝑎₀𝑎₁𝑎₂𝑎₃𝑎₄⋮=𝑏₀𝑏₁𝑏₂𝑏₃𝑏₄⋮, (23) where the generating functions of the column vectors are given, respectively, by 𝐴(𝑥) and 𝐵(𝑥). Then, equation of (23) is true if and only if the following equation holds: 𝑔(𝑥)∗_𝐹𝐴𝑓(𝑥)=𝐵(𝑥). (24)

Proof. Let 𝑔(𝑥),𝑓(𝑥)∈ℱ with 𝑔(0)≠0 and 𝑓(0)=0. Then, it can be written as 𝑔(𝑥)=𝑡(1−𝑥)^𝑛_𝐹,𝑓(𝑥)=𝑢𝑥(1−𝑥)^𝑚_𝐹, (25) where 𝑚,𝑛≥0 and 𝑡,𝑢∈ℝ. In this case, the matrix turns to ∣⋯∣∣𝑔𝑔∗_𝐹𝑓^[1]𝑔∗_𝐹𝑓^[2]𝑎∣∣∣∣∣∣∣∣∣⋮⋱⁰𝑎₁𝑎₂𝑎₃𝑎₄⋮ (26) and then𝑡𝑛−1𝑛−1_𝐹𝑡𝑛000⋯𝑛−1_𝐹𝑡𝑢𝑛+𝑚−2𝑛−1_𝐹𝑡00𝑛+1𝑛−1_𝐹𝑡𝑢𝑛+𝑚−1𝑛−1_𝐹𝑡𝑢²𝑛+2𝑚−3𝑛−1_𝐹0𝑡𝑛+2𝑛−1_𝐹𝑡𝑢𝑛+𝑚𝑛−1_𝐹𝑡𝑢²𝑛+2𝑚−2𝑛−1_𝐹𝑡𝑢³𝑛+3𝑚−4𝑛−1_𝐹𝑎⋮⋮⋮⋱⁰𝑎₁𝑎₂𝑎₃𝑎₄⋮=𝑎₀𝑡𝑛−1𝑛−1_𝐹𝑎₀𝑡𝑛𝑛−1_𝐹+𝑎₁𝑡𝑢𝑛+𝑚−2𝑛−1_𝐹𝑎₀𝑡𝑛+1𝑛−1_𝐹+𝑎₁𝑡𝑢𝑛+𝑚−1𝑛−1_𝐹+𝑎₂𝑡𝑢²𝑛+2𝑚−3𝑛−1_𝐹𝑎₀𝑡𝑛+2𝑛−1_𝐹+𝑎₁𝑡𝑢𝑛+𝑚𝑛−1_𝐹+𝑎₂𝑡𝑢²𝑛+2𝑚−2𝑛−1_𝐹+𝑎₃𝑡𝑢³𝑛+3𝑚−4𝑛−1_𝐹⋮ (27)and this yields 𝑎₀𝑡𝑛−1𝑛−1_𝐹+𝑎₀𝑡𝑛𝑛−1_𝐹+𝑎₁𝑡𝑢𝑛+𝑚−2𝑛−1_𝐹𝑥+𝑎₀𝑡𝑛+1𝑛−1_𝐹+𝑎₁𝑡𝑢𝑛+𝑚−1𝑛−1_𝐹+𝑎₂𝑡𝑢²𝑛+2𝑚−3𝑛−1_𝐹𝑥²+⋯=𝑎₀𝑡𝑛−1𝑛−1_𝐹𝑛+𝑡𝑛−1_𝐹𝑥+𝑡𝑛+1𝑛−1_𝐹𝑥²+⋯+𝑎₁𝑡𝑢𝑛+𝑚−2𝑛−1_𝐹𝑥+𝑡𝑢𝑛+𝑚−1𝑛−1_𝐹𝑥²+⋯+𝑎₂𝑡𝑢²𝑛+2𝑚−3𝑛−1_𝐹𝑥²+𝑡𝑢²𝑛+2𝑚−2𝑛−1_𝐹𝑥³+⋯+⋯=𝑎₀𝑔(𝑥)+𝑎₁𝑔(𝑥)∗_𝐹𝑓(𝑥)+𝑎₂𝑔(𝑥)∗_𝐹𝑓(𝑥)^[2]+𝑎₃𝑔(𝑥)∗_𝐹𝑓(𝑥)^[3](𝑥)∗+⋯=𝑔_𝐹𝑎₀+𝑎₁𝑓(𝑥)+𝑎₂𝑓(𝑥)^[2]+𝑎₃𝑓(𝑥)^[3](𝑥)∗+⋯=𝑔_𝐹𝐴𝑓(𝑥)=𝐵(𝑥) (28) and we have our result.

4. The 𝐹-Analogue of Riordan Representation of Pascal Matrices via Fibonomial Coefficients

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 𝐹-analogue of Riordan representation of 𝒫.

Theorem 6. Let 𝒫(𝑛,𝑚) be the infinite Pascal matrix via Fibonomial coefficients as in (5). Then, the 𝐹-analogue of Riordan representation of 𝒫(𝑛,𝑚) is given by 𝒫()=𝑡𝑛,𝑚(1−𝑥)^𝑛_𝐹,𝑢𝑥(1−𝑥)^𝑚_𝐹𝐹. (29)

Proof. We consider the infinite Pascal matrix via Fibonomial coefficients 𝒫(𝑛,𝑚). The entries of the 𝑗th column are given by 𝑝_𝑖𝑗=𝑥^𝑖𝑡𝑢^𝑗𝑛+𝑗𝑚−1+𝑖−𝑗𝑛+𝑗𝑚−1_𝐹𝑖=0,1,2,…. (30) Let 𝑔(𝑥)=𝑡(1−𝑥)^𝑛_𝐹,𝑓(𝑥)=𝑢𝑥(1−𝑥)^𝑚_𝐹; (31) then 𝑔(𝑥)∗_𝐹𝑓(𝑥)^[𝑗]=𝑡(1−𝑥)^𝑛_𝐹∗_𝐹𝑢𝑥(1−𝑥)^𝑚_𝐹^[𝑗]. (32) Using (22), we obtain 𝑔(𝑥)∗_𝐹𝑓(𝑥)^[𝑗]=𝑡𝑢^𝑗𝑥^𝑗(1−𝑥)_𝐹^{𝑛+𝑚𝑗}. (33) Taking (17) into account, we have 𝑔(𝑥)∗_𝐹𝑓(𝑥)^[𝑗]=𝑡𝑢^𝑗𝑥^𝑗(1−𝑥)_𝐹^{𝑛+𝑚𝑗}=_𝑖≥0𝑡𝑢^𝑗𝑛+𝑗𝑚−1+𝑖−𝑗𝑛+𝑗𝑚−1_𝐹𝑥^𝑖. (34) This proves that the generating function of the 𝑗th column of 𝒫(𝑛,𝑚) is 𝑡𝑢^𝑗𝑥^𝑗(1−𝑥)_𝐹^{𝑛+𝑚𝑗}. (35) In conclusion, the 𝐹-analogue of Riordan representation of 𝒫(𝑛,𝑚) is 𝑡(1−𝑥)^𝑛_𝐹,𝑢𝑥(1−𝑥)^𝑚_𝐹𝐹. (36)

Corollary 7. Let 𝒫 be in (6). Then, the 𝐹-analogue of Riordan representation of 𝒫 is given by 1𝒫=(1−𝑥)_𝐹,𝑥(1−𝑥)_𝐹𝐹. (37)

Corollary 8. Let 𝑈 be the Pascal matrix via Fibonomial coefficients of the first kind. Then, the 𝐹-analogue of Riordan representation of 𝑈 is given by 1(1−𝑥𝑡)_𝐹,𝑡(1−𝑥𝑡)_𝐹𝐹=𝑥1000000…𝑥100000…₂𝑥𝑥10000…₃2𝑥²𝑥2𝑥1000…₄3𝑥³6𝑥²𝑥3𝑥100…₅5𝑥⁴15𝑥³15𝑥²𝑥5𝑥10…₆8𝑥⁵40𝑥⁴60𝑥³40𝑥².8𝑥1…⋮⋮⋮⋮⋮⋮⋮⋱ (38)

Corollary 9. Let 𝑀 be the Pascal matrix via Fibonomial coefficients of the second kind. Then, the 𝐹-analogue of Riordan representation of 𝑀 is given by 1(1−𝑥𝑡)_𝐹,𝑥²𝑡(1−𝑥𝑡)_𝐹𝐹=100000…𝑥𝑥²𝑥0000…₂𝑥³𝑥⁴𝑥000…₃2𝑥⁴2𝑥⁵𝑥⁶𝑥00…₄3𝑥⁵6𝑥⁶3𝑥⁷𝑥⁸𝑥0…₅5𝑥⁶15𝑥⁷15𝑥⁸5𝑥⁹𝑥¹⁰….⋮⋮⋮⋮⋮⋮⋱ (39)

Corollary 10. Let Φ be the extended generalized Pascal matrix via Fibonomial coefficients. Then, 𝐹-analogue of the Riordan representation of Φ is 11−𝑥𝑦𝑡_𝐹,𝑦²𝑡1−𝑥𝑦𝑡_𝐹𝐹=10000…𝑥𝑦𝑦²𝑥000…₂𝑦²𝑥𝑦³𝑦⁴𝑥00…₃𝑦³2𝑥²𝑦⁴2𝑥𝑦⁵𝑦⁶𝑥0…₄𝑦⁴3𝑥³𝑦⁵6𝑥²𝑦⁶3𝑥𝑦⁷𝑦⁸….⋮⋮⋮⋮⋮⋱ (40)

Conflict of Interests

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