On the (p, q)–Chebyshev Polynomials and Related Polynomials

Kızılateş, Can; Tuglu, Naim; Çekіm, Bayram

doi:10.3390/math7020136

Cited by 11 publications

(8 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We provide the generating function for ( , ) k P s p in the following theorem. Here the functions () fx  defined in [9] with  operator and, similarly, 2 () fx  with the same operator defined in [14], will be a guide for the proof.…”

Section: mentioning

confidence: 99%

See 1 more Smart Citation

p-Analogue of biperiodic Pell and Pell–Lucas polynomials

Kuloğlu¹,

Özkan²,

Shannon³

2023

NNTDM

View full text Add to dashboard Cite

In this study, a binomial sum, unlike but analogous to the usual binomial sums, is expressed with a different definition and termed the p-integer sum. Based on this definition, p-analogue Pell and Pell–Lucas polynomials are established and the generating functions of these new polynomials are obtained. Some theorems and propositions depending on the generating functions are also expressed. Then, by association with these, the polynomials of so-called ‘incomplete’ number sequences have been obtained, and elegant summation relations provided. The paper has also been placed in the appropriate historical context for ease of further development.

show abstract

Section: mentioning

confidence: 99%

“…with initial conditions 01 1. QQ  Incomplete and biperiodic studies have been discussed by many authors [1,4,[9][10][11][12][13][14]19]. Depending on k any integer, incomplete recurrences of the k-Pell and k-Pell-Lucas sequence [3] are defined as follows:…”

Section: Introductionmentioning

confidence: 99%

p-Analogue of biperiodic Pell and Pell–Lucas polynomials

Kuloğlu¹,

Özkan²,

Shannon³

2023

NNTDM

View full text Add to dashboard Cite

show abstract

“…The theory of ðp, qÞ-calculus or post quantum calculus has recently been applied in many areas of mathematics, physics and engineering, such as biology, mechanics, economics, electrochemistry, probability theory, approximation theory, statistics, number theory, quantum theory, theory of relativity, and statistical mechanics, etc. For more details on this topic ðp, qÞ-calculus, see, for example, [1][2][3][4][5][6]. Burban and Klimyk [3], Duran et al [7][8][9][10], Jagannathan [11], Jagannathan and Srinivasa [12], Sahai and Yadav [13] have earlier investigated some properties of the two parameter quantum calculus.…”

Section: Introductionmentioning

confidence: 99%

On the p,q-Humbert Functions from the View Point of the Generating Function Method

Shehata

2020

Journal of Function Spaces

View full text Add to dashboard Cite

The main object of the present paper is to construct new p,q-analogy definitions of various families of p,q-Humbert functions using the generating function method as a starting point. This study shows a class of several results of p,q-Humbert functions with the help of the generating functions such as explicit representations and recurrence relations, especially differential recurrence relations, and prove some of their significant properties of these functions.

show abstract

“…The Fibonacci sequence was generalized in many different ways, some of them you can find in [6][7][8][9][10][11][12][13][14][15][16][17][18]. By keeping its order (which is 2), we have a general Lucas sequence (C n ) n = (C n (a, b)) n which is defined by the recurrence C n = aC n−1 + bC n−2 , for n ≥ 2, and with C i = i, for i ∈ {0, 1} (moreover, the integer parameters a and b must be such that if x 2 − ax − b = (x − r)(x − s), then r/s is not a root of unity).…”

Section: Introductionmentioning

confidence: 99%

Some Diophantine Problems Related to k-Fibonacci Numbers

Trojovský

Hubálovský

2020

Mathematics

View full text Add to dashboard Cite

Let k ≥ 1 be an integer and denote ( F k , n ) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence ( L k , n ) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences ( F k , n ) n ≥ 0 and ( L k , n ) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ ( F k , m ) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ ( k F k , m ) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k .

show abstract

On the (p, q)–Chebyshev Polynomials and Related Polynomials

Cited by 11 publications

References 18 publications

p-Analogue of biperiodic Pell and Pell–Lucas polynomials

p-Analogue of biperiodic Pell and Pell–Lucas polynomials

On the p,q-Humbert Functions from the View Point of the Generating Function Method

Some Diophantine Problems Related to k-Fibonacci Numbers

Contact Info

Product

Resources

About