A total variable-order variation model for image denoising

Abdelilah Hakim; Anouar Ben-Loghfyry; Abdelilah Hakim; Anouar Ben-Loghfyry

doi:10.3934/math.2019.5.1320

AIMS Mathematics

2019, Volume 4, Issue 5: 1320-1335. doi: 10.3934/math.2019.5.1320

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Research article

A total variable-order variation model for image denoising

Abdelilah Hakim ,
Anouar Ben-Loghfyry ^,

LAMAI laboratory, university of Cadi Ayyad, Faculty of sciences and technology, Marrakesh, Morocco

Received: 06 March 2019 Accepted: 14 August 2019 Published: 04 September 2019
MSC : 26A33, 94A08

In this paper, we explore a new variational model based on the fractional derivative and total variation.Due to some metrics, our approach shows great results compared to other competitive models.In particular, deleting the noise and preserving edges, features and corners are headlights to our approach.For the fractional variable-order derivatives, different discretizations were presented to comparison.The theoretical results are validated by the Primal Dual Projected Gradient (PDPG) Algorithm which is well adapted to the fractional calculus.

Keywords:

Citation: Abdelilah Hakim, Anouar Ben-Loghfyry. A total variable-order variation model for image denoising[J]. AIMS Mathematics, 2019, 4(5): 1320-1335. doi: 10.3934/math.2019.5.1320

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Abstract

1. Introduction

Special polynomials, generating functions, and the trigonometric functions are used not only in mathematics but also in many branches of science such as statistics, mathematical physics, and engineering.

Let $\mathbb{N}, \mathbb{Z}, \mathbb{R}$ and $\mathbb{C}$ indicate the set of positive integers, the set of integers, the set of real numbers, and the set of complex numbers, respectively. Let $\alpha \in \mathbb{N} _{0} = \mathbb{N} \cup \left\{ 0\right\}$ and $\lambda \in \mathbb{C}$ (or $\mathbb{R}).$

The Apostol-Bernoulli polynomials $\mathcal{B}_{n}^{(\alpha)}(x; \lambda)$ of order $\alpha$ are defined by means of the following exponential generating function (see ^[1,2,3]):

$\begin{equation} \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n}^{(\alpha )}(x;\lambda )\frac{t^{n}}{n!} = \left( \frac{t}{\lambda e^{t}-1}\right) ^{\alpha }e^{xt}, \end{equation}$

(1.1)

$\begin{equation} (\lambda \in \mathbb{C} ;\text{ }\left\vert t\right\vert < 2\pi , \text{ when }\lambda = 1;\text{ and } \left\vert t\right\vert < \left\vert \log \lambda \right\vert , \text{ when } \lambda \neq 1). \end{equation}$

(1.2)

Note that $\mathcal{B}_{n}^{(\alpha)}(x; 1) = \mathcal{B}_{n}^{(\alpha)}(x)$ denote the Bernoulli polynomials of order $\alpha$ and $\mathcal{B} _{n}^{(\alpha)}(0;\lambda) = \mathcal{B}_{n}^{(\alpha)}(\lambda)$ denote the Apostol-Bernoulli numbers of order $\alpha,$ respectively. Setting $\alpha = 1$ into (1.1), we get $\mathcal{B}_{n}^{(1)}(\lambda) = \mathcal{B}_{n}(\lambda)$ which are the so-called Apostol-Bernoulli numbers.

The Apostol-Euler polynomials $\mathcal{E}_{n}^{(\alpha)}(x; \lambda)$ of order $\alpha$ are defined by means of the following exponential generating function (see ^[4,5]):

$\begin{equation} \sum\limits_{n = 0}^{\infty }\mathcal{E}_{n}^{(\alpha )}(x;\lambda )\frac{t^{n}}{n!} = \left( \frac{2}{\lambda e^{t}+1}\right) ^{\alpha }e^{xt}, \end{equation}$

(1.3)

$\begin{equation} \left( \left\vert t\right\vert < \pi \text{ when }\lambda = 1;\text{ } \left\vert t\right\vert < |\log (-\lambda )|\text{ when }\lambda \neq 1;1^{\alpha }: = 1\right). \end{equation}$

(1.4)

By virtue of (1.3), we have $\mathcal{E}_{n}^{(\alpha)}(x; 1) = \mathcal{E}_{n}^{(\alpha)}(x)$ denote the Euler polynomials of order $\alpha$ and $\mathcal{E}_{n}^{(\alpha)}(0;\lambda) = \mathcal{E} _{n}^{(\alpha)}(\lambda)$ denote the Apostol-Euler numbers of order $\alpha,$ respectively. Setting $\alpha = 1$ into (1.3), we get $\mathcal{E}_{n}^{(1)}(\lambda) = \mathcal{E}_{n}(\lambda)$ which are the so-called Apostol-Euler numbers.

The Apostol-Genocchi polynomials $\mathcal{G}_{n}^{(\alpha)}(x; \lambda)$ of order $\alpha$ are defined by means of the following exponential generating function (see ^[6]):

$\begin{equation} \sum\limits_{n = 0}^{\infty }\mathcal{G}_{n}^{(\alpha )}(x;\lambda )\frac{t^{n}}{n!} = \left( \frac{2t}{\lambda e^{t}+1}\right) ^{\alpha }e^{xt}, \end{equation}$

(1.5)

$\begin{equation} \left( |t| < \pi \text{ when }\lambda = 1;\text{ }\left\vert t\right\vert < |\log (-\lambda )|\text{ when }\lambda \neq 1;1^{\alpha }: = 1\right). \end{equation}$

(1.6)

By virtue of (1.5), we have $\mathcal{G}_{n}^{(\alpha)}(x; 1) = \mathcal{G}_{n}^{(\alpha)}(x)$ denote the Genocchi polynomials of order $\alpha$ and $\mathcal{G}_{n}^{(\alpha)}(0;\lambda) = \mathcal{G} _{n}^{(\alpha)}(\lambda)$ denote the Apostol-Genocchi numbers of order $\alpha,$ respectively. Setting $\alpha = 1$ into (1.5), we get $\mathcal{G}_{n}^{(1)}(\lambda) = \mathcal{G}_{n}(\lambda)$ which are the so-called Apostol-Genocchi numbers.

In recent years, many generalizations of these polynomials have been studied by mathematicians. See for example ^{[7,8,9,10,11,12,13,14,15,16,17,18,19,20]}. With the aid of these polynomials two parametric kinds of Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi of order $\alpha$ defined by Srivastava et al. ^[15,19] whose generating functions are given by

$\begin{equation} \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n}^{(c, \alpha )}(x, y;\lambda )\frac{t^{n}}{ n!} = \left( \frac{t}{\lambda e^{t}-1}\right) ^{\alpha }e^{xt}\cos (yt), \end{equation}$

(1.7)

$\begin{equation} \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n}^{(s, \alpha )}(x, y;\lambda )\frac{t^{n}}{ n!} = \left( \frac{t}{\lambda e^{t}-1}\right) ^{\alpha }e^{xt}\sin (yt), \end{equation}$

(1.8)

and

$\begin{equation} \sum\limits_{n = 0}^{\infty }\mathcal{E}_{n}^{(c, \alpha )}(x, y;\lambda )\frac{t^{n}}{ n!} = \left( \frac{2}{\lambda e^{t}+1}\right) ^{\alpha }e^{xt}\cos (yt), \end{equation}$

(1.9)

$\begin{equation} \sum\limits_{n = 0}^{\infty }\mathcal{E}_{n}^{(s, \alpha )}(x, y;\lambda )\frac{t^{n}}{ n!} = \left( \frac{2}{\lambda e^{t}+1}\right) ^{\alpha }e^{xt}\sin (yt), \end{equation}$

(1.10)

and

$\begin{equation} \sum\limits_{n = 0}^{\infty }\mathcal{G}_{n}^{(c, \alpha )}(x, y;\lambda )\frac{t^{n}}{ n!} = \left( \frac{2t}{\lambda e^{t}+1}\right) ^{\alpha }e^{xt}\cos (yt), \end{equation}$

(1.11)

$\begin{equation} \sum\limits_{n = 0}^{\infty }\mathcal{G}_{n}^{(s, \alpha )}(x, y;\lambda )\frac{t^{n}}{ n!} = \left( \frac{2t}{\lambda e^{t}+1}\right) ^{\alpha }e^{xt}\sin (yt). \end{equation}$

(1.12)

Remark 1.1. Note that the symbols $c$ and $s$ occurring in the superscripts on the left-hand sides of these last Eqs (1.7)–(1.12) indicate the presence of the trigonometric cosine and the trigonometric sine functions, respectively, in the generating functions on the corresponding right-hand sides.

The motivation of this paper is to obtain $F$ -analogues of the Eqs (1.7)–(1.12) with the help of the Golden calculus. Namely, we define the parametric Apostol Bernoulli-Fibonacci, the Apostol Euler-Fibonacci, and the Apostol Genocchi-Fibonacci polynomials by means of the Golden Calculus. Utilizing the Golden-Euler formula and these generating functions with their functional equations, numerous properties of these polynomials are given. The special cases of these polynomials and numbers are studied in detail. The rest of this paper is structured as follows. In Section 2, we present some key definitions and properties that are crucial to Golden calculus. Then, with the help of the Golden calculus, we mention some polynomials that have been previously defined in the literature. In Section 3, considering the properties of Golden calculus, we introduce six families of two-parameter polynomials with the help of Golden trigonometric functions and exponential functions. Then, in the three subsections of this section, we examine the various properties of these polynomials defined with the help of generating functions and their functional equations.

2. Golden calculus

In this part of the our paper, we mention some definitions and properties related to Golden calculus (or $F$ -calculus).

The Fibonacci sequence is defined by means of the following recurrence relation:

$\begin{equation*} F_{n} = F_{n-1}+F_{n-2}, { \ \ }n\geq 2 \end{equation*}$

where $F_{0} = 0,$ $F_{1} = 1.$ Fibonacci numbers can be expressed explicitly as

$\begin{equation*} F_{n} = \frac{\alpha ^{n}-\beta ^{n}}{\alpha -\beta }, \end{equation*}$

where $\alpha = \frac{1+\sqrt{5}}{2}$ and $\beta = \frac{1-\sqrt{5}}{2}.$ $\alpha \approx 1, 6180339\ldots$ is called Golden ratio. The golden ratio is frequently used in many branches of science as well as mathematics. Interestingly, this mysterious number also appears in architecture and art. Miscellaneous properties of Golden calculus have been defined and studied in detail by Pashaev and Nalci ^[21]. Therefore, ^[21] is the key reference for Golden calculus. In addition readers can also refer to Pashaev ^[22], Krot ^[23], and Ozvatan ^[24].

The product of Fibonacci numbers, called $F$ -factorial was defined as follows:

$\begin{equation} F_{1}F_{2}F_{3}\ldots F_{n} = F_{n}!, \end{equation}$

(2.1)

where $F_{0}! = 1.$ The binomial theorem for the $F$ -analogues (or-Golden binomial theorem) are given by

$\begin{equation} \left( x+y\right) _{F}^{n} = \sum\limits_{k = 0}^{n}\left( -1\right) ^{\tbinom{k}{2}} \dbinom{n}{k}_{F}x^{n-k}y^{k}, \end{equation}$

(2.2)

in terms of the Golden binomial coefficients, called as Fibonomials

$\begin{equation*} \dbinom{n}{k}_{F} = \frac{F_{n}!}{F_{n-k}!F_{k}!} \end{equation*}$

with $n$ and $k$ being nonnegative integers, $n\geq k.$ Golden binomial coefficients (or-Fibonomial coefficients) satisfy the following identities as follows:

$\begin{equation*} \dbinom{n}{k}_{F} = \beta ^{k}\dbinom{n-1}{k}_{F}+\alpha^{n-k} \dbinom{n-1}{k-1}_{F}, \end{equation*}$

and

$\begin{equation*} \dbinom{n}{k}_{F} = \alpha ^{k}\dbinom{n-1}{k}_{F}+\beta ^{n-k}\dbinom{n-1}{ k-1 }_{F}. \end{equation*}$

The Golden derivative defined as follows:

$\begin{equation} \frac{\partial _{F}}{\partial _{F}x}\left( f(x)\right) = \frac{f\left( \alpha x\right) -f\left( -\frac{x}{\alpha }\right) }{\left( \alpha -\left( -\frac{1 }{\alpha }\right) \right) x} = \frac{f\left( \alpha x\right) -f\left( \beta x\right) }{\left( \alpha -\beta \right) x}. \end{equation}$

(2.3)

The Golden Leibnitz rule and the Golden derivative of the quotient of $f(x)$ and $g(x)$ can be given as

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}x}\left( f(x)g(x)\right) = \frac{\partial _{F}}{\partial _{F}x}f(x)g(\alpha x)+f(-\frac{x}{\alpha })\frac{\partial _{F} }{\partial _{F}x}g(x), \end{equation*}$

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}x}\left( \frac{f(x)}{g(x)}\right) = \frac{ \frac{\partial _{F}}{\partial _{F}x}f(x)g(\alpha x)-\frac{\partial _{F}}{ \partial _{F}x}g(x)f\left( \alpha x\right) }{g(\alpha x)g(\frac{-x}{\alpha } ) }, \end{equation*}$

respectively. The first and second type of Golden exponential functions are defined as

$\begin{equation} e_{F}^{x} = \sum\limits_{n = 0}^{\infty }\frac{\left( x\right) _{F}^{n}}{F_{n}!}, \end{equation}$

(2.4)

and

$\begin{equation} E_{F}^{x} = \sum\limits_{n = 0}^{\infty }\left( -1\right) ^{\tbinom{n}{2}}\frac{\left( x\right) _{F}^{n}}{F_{n}!}. \end{equation}$

(2.5)

Briefly, we use the following notations throughout the paper

$\begin{equation} e_{F}^{x} = \sum\limits_{n = 0}^{\infty }\frac{x^{n}}{F_{n}!}, \end{equation}$

(2.6)

and

$\begin{equation} E_{F}^{x} = \sum\limits_{n = 0}^{\infty }\left( -1\right) ^{\tbinom{n}{2}}\frac{x^{n}}{ F_{n}!}. \end{equation}$

(2.7)

Using the Eqs (2.2), (2.4), and (2.5), the following equation can be given

$\begin{equation} e_{F}^{x}E_{F}^{y} = e_{F}^{(x+y)_{F}}. \end{equation}$

(2.8)

The Fibonacci cosine and sine (Golden trigonometric functions) are defined by the power series as

$\begin{equation} \cos _{F}\left( x\right) = \sum\limits_{n = 0}^{\infty }\left( -1\right) ^{n}\frac{ x^{2n}}{F_{2n}!}, \end{equation}$

(2.9)

and

$\begin{equation} \sin _{F}\left( x\right) = \sum\limits_{n = 0}^{\infty }\left( -1\right) ^{n}\frac{ x^{2n+1}}{F_{2n+1}!}. \end{equation}$

(2.10)

For arbitrary number $k$ , Golden derivatives of $e_{F}^{kx},$ $E_{F}^{kx},$ $\cos _{F}\left(kx\right),$ and $\sin _{F}\left(kx\right)$ functions are

$\begin{equation} \frac{\partial _{F}}{\partial _{F}x}\left( e_{F}^{kx}\right) = ke_{F}^{kx}, \end{equation}$

(2.11)

$\begin{equation} \frac{\partial _{F}}{\partial _{F}x}\left( E_{F}^{kx}\right) = kE_{F}^{-kx}, \end{equation}$

(2.12)

$\begin{equation} \frac{\partial _{F}}{\partial _{F}x}\left( \cos _{F}\left( kx\right) \right) = -k\sin _{F}\left( kx\right) , \end{equation}$

(2.13)

and

$\begin{equation} \frac{\partial _{F}}{\partial _{F}x}\left( \sin _{F}\left( x\right) \right) = k\cos _{F}\left( kx\right) . \end{equation}$

(2.14)

Using (2.4), Pashaev and Ozvatan ^[25] defined the Bernoulli-Fibonacci polynomials and related numbers. After that Kus et al. ^[26] introduced the Euler-Fibonacci numbers and polynomials. Moreover they gave some identities and matrix representations for Bernoulli-Fibonacci polynomials and Euler-Fibonacci polynomials. Very recently, Tuglu and Ercan ^[27] (also, ^[28]) defined the generalized Bernoulli-Fibonacci polynomials and generalized Euler-Fibonacci polynomials, namely, they studied the Apostol Bernoulli-Fibonacci and Apostol Euler-Fibonacci of order $\alpha$ as follows:

$\begin{equation*} \left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{xt} = \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{\alpha }\left( x;\lambda \right) \frac{t^{n}}{F_{n}!}, \end{equation*}$

and

$\begin{equation*} \left( \frac{2}{\lambda e_{F}^{t}+1}\right) ^{\alpha }e_{F}^{xt} = \sum\limits_{n = 0}^{\infty }\mathcal{E}_{n, F}^{\alpha }\left( x;\lambda \right) \frac{t^{n}}{F_{n}!}. \end{equation*}$

3. Generalized parametric types of Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials via Golden calculus

Krot ^[23] defined the fibonomial convolution of two sequences as follows. Let $a_{n}$ and $b_{n}$ are two sequences with the following generating functions

$\begin{equation*} A_{F}(t) = \sum\limits_{n = 0}^{\infty }a_{n}\frac{t^{n}}{F_{n}!}\text{ and } B_{F}(t) = \sum\limits_{n = 0}^{\infty }b_{n}\frac{t^{n}}{F_{n}!}, \end{equation*}$

then their fibonomial convolution is defined as

$\begin{equation*} c_{n} = a_{n}\ast b_{n} = \sum\limits_{l = 0}^{n}\binom{n}{k}_{F}a_{l}b_{n-l}. \end{equation*}$

So, the generating function takes the form

$\begin{equation*} C_{F}(t) = A_{F}(t)B_{F}(t) = \sum\limits_{n = 0}^{\infty }c_{n}\frac{t^{n}}{F_{n}!}. \end{equation*}$

Let $p, q\in \mathbb{R}.$ The Taylor series of the functions $e_{F}^{pt}\cos _{F}\left(qt\right)$ and $e_{F}^{pt}\sin _{F}\left(qt\right)$ can be express as follows:

$\begin{equation} e_{F}^{pt}\cos _{F}\left( qt\right) = \sum\limits_{n = 0}^{\infty }\mathcal{C} _{n, F}\left( p, q\right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.1)

and

$\begin{equation} e_{F}^{pt}\sin _{F}\left( qt\right) = \sum\limits_{n = 0}^{\infty }\mathcal{S} _{n, F}\left( p, q\right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.2)

where

$\begin{equation} \mathcal{C}_{n, F}\left( p, q\right) = \sum\limits_{k = 0}^{\left\lfloor \frac{n}{2} \right\rfloor }\left( -1\right) ^{k}\dbinom{n}{2k}_{F}\text{ }p^{n-2k}q^{2k}, \end{equation}$

(3.3)

$\begin{equation} \mathcal{S}_{n, F}\left( p, q\right) = \sum\limits_{k = 0}^{\left\lfloor \frac{n-1}{2} \right\rfloor }\left( -1\right) ^{k}\dbinom{n}{2k+1}_{F}\text{ } p^{n-2k-1}q^{2k+1}. \end{equation}$

(3.4)

By virtue of above definitions of $\mathcal{C}_{n, F}\left(p, q\right)$ and $\mathcal{S}_{n, F}\left(p, q\right)$ and the numbers $\mathcal{B} _{n, F}^{\left(\alpha \right) }\left(\lambda \right),$ $\mathcal{E} _{n, F}^{\left(\alpha \right) }\left(\lambda \right)$ and $\mathcal{G} _{n, F}^{\left(\alpha \right) }\left(\lambda \right)$ , we can define two parametric types of the Apostol Bernoulli-Fibonacci polynomials, the Apostol Euler-Fibonacci polynomials, and the Apostol Genocchi-Fibonacci polynomials of order $\alpha,$ as follows:

$\begin{equation*} \mathcal{B}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) = \mathcal{B} _{n, F}^{\left( \alpha \right) }\left( \lambda \right) \ast \mathcal{C} _{n, F}\left( p, q\right) , \end{equation*}$

$\begin{equation*} \mathcal{B}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) = \mathcal{B} _{n, F}^{\left( \alpha \right) }\left( \lambda \right) \ast \mathcal{S} _{n, F}\left( p, q\right) , \end{equation*}$

$\begin{equation*} \mathcal{E}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) = \mathcal{E} _{n, F}^{\left( \alpha \right) }\left( \lambda \right) \ast \mathcal{C} _{n, F}\left( p, q\right) , \end{equation*}$

$\begin{equation*} \mathcal{E}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) = \mathcal{E} _{n, F}^{\left( \alpha \right) }\left( \lambda \right) \ast \mathcal{S} _{n, F}\left( p, q\right) , \end{equation*}$

$\begin{equation*} \mathcal{G}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) = \mathcal{G} _{n, F}^{\left( \alpha \right) }\left( \lambda \right) \ast \mathcal{C} _{n, F}\left( p, q\right) , \end{equation*}$

$\begin{equation*} \mathcal{G}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) = \mathcal{G} _{n, F}^{\left( \alpha \right) }\left( \lambda \right) \ast \mathcal{S} _{n, F}\left( p, q\right) , \end{equation*}$

whose exponential generating functions are given, respectively, by

$\begin{equation} \left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{pt}\cos _{F}\left( qt\right) = \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.5)

$\begin{equation} \left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{pt}\sin _{F}\left( qt\right) = \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.6)

$\begin{equation} \left( \frac{2}{\lambda e_{F}^{t}+1}\right) ^{\alpha }e_{F}^{pt}\cos _{F}\left( qt\right) = \sum\limits_{n = 0}^{\infty }\mathcal{E}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.7)

$\begin{equation} \left( \frac{2}{\lambda e_{F}^{t}+1}\right) ^{\alpha }e_{F}^{pt}\sin _{F}\left( qt\right) = \sum\limits_{n = 0}^{\infty }\mathcal{E}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.8)

$\begin{equation} \left( \frac{2t}{\lambda e_{F}^{t}+1}\right) ^{\alpha }e_{F}^{pt}\cos _{F}\left( qt\right) = \sum\limits_{n = 0}^{\infty }\mathcal{G}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.9)

$\begin{equation} \left( \frac{2t}{\lambda e_{F}^{t}+1}\right) ^{\alpha }e_{F}^{pt}\sin _{F}\left( qt\right) = \sum\limits_{n = 0}^{\infty }\mathcal{G}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!}. \end{equation}$

(3.10)

Remark 3.1. By virtue of (3.5) and (3.6), when $\lambda \neq 1$ , $\mathcal{ B}_{0, F}^{(c, \alpha)}\left(p, q; \lambda \right) = 0$ and when $\lambda = 1,$ $\mathcal{B}_{0, F}^{(c, \alpha)}\left(p, q; 1\right) = 1.$ Moreover for $\forall \lambda \in \mathbb{C},$ $\mathcal{B}_{0, F}^{(s, \alpha)}\left(p, q; \lambda \right) = 0.$

Remark 3.2. By virtue of (3.7), when $\lambda = -1$ , $\mathcal{E}_{0, F}^{(c, \alpha)}\left(p, q; -1\right)$ is undefined and when $\lambda \neq -1,$ $\mathcal{E }_{0, F}^{(c, \alpha)}\left(p, q; \lambda \right) = \left(\frac{2}{\lambda +1} \right) ^{\alpha }.$ Also, from (3.8), when $\lambda \neq -1,$ $\mathcal{E}_{0, F}^{(s, \alpha)}\left(p, q; \lambda \right) = 0.$ For $\lambda = -1,$ $\mathcal{E}_{0, F}^{(s, \alpha)}\left(p, q; -1\right)$ is determined according to the values of $\alpha.$

Remark 3.3. By virtue of (3.9) and (3.10), when $\lambda \neq -1$ , $\mathcal{G}_{0, F}^{(c, \alpha)}\left(p, q; \lambda \right) = 0$ and $\lambda = -1,$ $\mathcal{G}_{0, F}^{(c, \alpha)}\left(p, q; -1\right) = \left(-2\right) ^{\alpha }.$ Moreover for $\forall \lambda \in \mathbb{C},$ $\mathcal{G}_{0, F}^{(s, \alpha)}\left(p, q; \lambda \right) = 0.$

Remark 3.4. If we take $\alpha = 1$ and $q = 0$ in (3.5), (3.7), and (3.9), we get Apostol Bernoulli-Fibonacci polynomials, Apostol Euler-Fibonacci polynomials, and Apostol Genocchi-Fibonacci polynomials

$\begin{equation} \frac{te_{F}^{pt}}{\lambda e_{F}^{t}-1} = \sum\limits_{n = 0}^{\infty }\mathcal{B} _{n, F}\left( p;\lambda \right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.11)

$\begin{equation} \frac{2e_{F}^{pt}}{\lambda e_{F}^{t}+1} = \sum\limits_{n = 0}^{\infty }\mathcal{E} _{n, F}\left( p;\lambda \right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.12)

$\begin{equation} \frac{2te_{F}^{pt}}{\lambda e_{F}^{t}+1} = \sum\limits_{n = 0}^{\infty }\mathcal{G} _{n, F}\left( p;\lambda \right) \frac{t^{n}}{F_{n}!}, \end{equation}$

(3.13)

respectively. If we take $p = 0$ in (3.11)–(3.13), we obtain Apostol Bernoulli-Fibonacci numbers $\mathcal{B}_{n, F}\left(\lambda \right)$ , Apostol Euler-Fibonacci numbers $\mathcal{E}_{n, F}\left(\lambda \right)$ , and Apostol Genocchi-Fibonacci numbers $\mathcal{G}_{n, F}\left(\lambda \right).$

3.1. Basic properties of $\mathcal{B}_{n, F}^{(c, \alpha)}\left(p, q; \lambda \right)$ and $\mathcal{B}_{n, F}^{(s, \alpha)}\left(p, q; \lambda \right)$

Theorem 3.1. The following identities hold true:

$\begin{equation} \mathcal{B}_{n, F}^{(c, \alpha )}\left( p+r, q;\lambda \right) = \sum\limits_{k = 0}^{n}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{ B }_{k, F}^{(c, \alpha )}\left( p, q;\lambda \right) r^{n-k}, \end{equation}$

(3.14)

and

$\begin{equation} \mathcal{B}_{n, F}^{(s, \alpha )}\left( p+r, q;\lambda \right) = \sum\limits_{k = 0}^{n}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{ B }_{k, F}^{(s, \alpha )}\left( p, q;\lambda \right) r^{n-k}. \end{equation}$

(3.15)

Proof. By applying (3.5), we first derive the following functional equation:

$\begin{eqnarray*} \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, \alpha )}\left( p+r, q;\lambda \right) \frac{t^{n}}{F_{n}!} & = &\left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{\left( p+r\right) _{F}t}\cos _{F}\left( qt\right) \\ & = &\left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{pt}\cos _{F}\left( qt\right) E_{F}^{rt}, \end{eqnarray*}$

which readily yields

$\begin{eqnarray*} \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, \alpha )}\left( p+r, q;\lambda \right) \frac{t^{n}}{F_{n}!} & = &\left( \sum\limits_{n = 0}^{\infty }\mathcal{B} _{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!}\right) \left( \sum\limits_{n = 0}^{\infty }\left( -1\right) ^{\tbinom{n}{2}}\frac{\left( rt\right) ^{n}}{F_{n}!}\right) \\ & = &\sum\limits_{n = 0}^{\infty }\left( \sum\limits_{k = 0}^{n}\left( -1\right) ^{\tbinom{n-k}{ 2 }}\dbinom{n}{k}_{F}\mathcal{B}_{k, F}^{(c, \alpha )}\left( p, q;\lambda \right) r^{n-k}\right) \frac{t^{n}}{F_{n}!}. \end{eqnarray*}$

Comparing the coefficients of $t^{n}$ on both sides of this last equation, we have

$\begin{equation*} \mathcal{B}_{n, F}^{(c, \alpha )}\left( p+r, q;\lambda \right) = \sum\limits_{k = 0}^{n}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{ B }_{k, F}^{(c, \alpha )}\left( p, q;\lambda \right) r^{n-k}, \end{equation*}$

which proves the result (3.14). The assertion (3.15) can be proved similarly.

Remark 3.5. We claim that

$\begin{equation*} \mathcal{B}_{n, F}^{(c, \alpha )}\left( p+1, q;\lambda \right) -\mathcal{B} _{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n-1}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{B}_{k, F}^{(c, \alpha )}\left( p, q;\lambda \right) , \end{equation*}$

and

$\begin{equation*} \mathcal{B}_{n, F}^{(s, \alpha )}\left( p+1, q;\lambda \right) -\mathcal{B} _{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n-1}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{B}_{k, F}^{(s, \alpha )}\left( p, q;\lambda \right) . \end{equation*}$

Theorem 3.2. For every $n\in \mathbb{N},$ following identities hold true:

$\begin{equation} \frac{\partial _{F}}{\partial _{F}p}\left\{ \mathcal{B}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \right\} = F_{n}\mathcal{B}_{n-1, F}^{(c, \alpha )}\left( p, q;\lambda \right) , \end{equation}$

(3.16)

$\begin{equation} \frac{\partial _{F}}{\partial _{F}p}\left\{ \mathcal{B}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) \right\} = F_{n}\mathcal{B}_{n-1, F}^{(s, \alpha )}\left( p, q;\lambda \right) , \end{equation}$

(3.17)

$\begin{equation} \frac{\partial _{F}}{\partial _{F}q}\left\{ \mathcal{B}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \right\} = -F_{n}\mathcal{B}_{n-1, F}^{(s, \alpha )}\left( p, q;\lambda \right) , \end{equation}$

(3.18)

and

$\begin{equation} \frac{\partial _{F}}{\partial _{F}q}\left\{ \mathcal{B}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) \right\} = F_{n}\mathcal{B}_{n-1, F}^{(c, \alpha )}\left( p, q;\lambda \right) . \end{equation}$

(3.19)

Proof. Using (3.5) and applying the Golden derivative operator $\frac{ \partial _{F}}{\partial _{F}p},$ we obtain

$\begin{eqnarray*} \sum\limits_{n = 0}^{\infty }\frac{\partial _{F}}{\partial _{F}p}\left\{ \mathcal{B} _{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \right\} \frac{t^{n}}{F_{n}!} & = &\frac{\partial _{F}}{\partial _{F}p}\left\{ \left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{pt}\right\} \cos _{F}\left( qt\right) \\ & = &t\left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{pt}\cos _{F}\left( qt\right) \\ & = &\sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \frac{t^{n+1}}{F_{n}!} \\ & = &\sum\limits_{n = 1}^{\infty }\mathcal{B}_{n-1, F}^{(c, \alpha )}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n-1}!}. \end{eqnarray*}$

By comparing the coefficients of $t^{n}$ on both sides of this last equation, we arrive at the desired result (3.16). To prove (3.18), using (3.5) and applying the Golden derivative operator $\frac{\partial _{F}}{\partial _{F}q},$ we find that

$\begin{eqnarray*} \sum\limits_{n = 0}^{\infty }\frac{\partial _{F}}{\partial _{F}q}\left\{ \mathcal{B} _{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \right\} \frac{t^{n}}{F_{n}!} & = &\frac{\partial _{F}}{\partial _{F}q}\left\{ \left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{pt}\cos _{F}\left( qt\right) \right\} \\ & = &\frac{\partial _{F}}{\partial _{F}q}\left\{ \cos _{F}\left( qt\right) \right\} \left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{pt} \\ & = &-t\sin _{F}\left( qt\right) \left( \frac{t}{\lambda e_{F}^{t}-1}\right) ^{\alpha }e_{F}^{pt} \\ & = &\sum\limits_{n = 1}^{\infty }-\mathcal{B}_{n-1, F}^{(s, \alpha )}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n-1}!}. \end{eqnarray*}$

Comparing the coefficients of $t^{n}$ on both sides of this last equation, we arrive at the desired result (3.18). Equations (3.17) and (3.19) can be similarly derived.

Theorem 3.3. The following identities hold true:

$\begin{equation} \mathcal{B}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}\dbinom{n }{k}_{F}\mathcal{B}_{k, F}\left( \lambda \right) \mathcal{C}_{n-k}\left( p, q\right) , \end{equation}$

(3.20)

and

$\begin{equation} \mathcal{B}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}\dbinom{n }{k}_{F}\mathcal{B}_{k, F}\left( \lambda \right) \mathcal{S}_{n-k}\left( p, q\right) . \end{equation}$

(3.21)

Proof. Setting $\alpha = 1$ in (3.5) and using (3.1), we find that

$\begin{eqnarray*} \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!} & = &\frac{t}{\lambda e_{F}^{t}-1}e_{F}^{pt}\cos _{F}\left( qt\right) \\ & = &\left( \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}\left( p;\lambda \right) \frac{t^{n}}{F_{n}!}\right) \left( \sum\limits_{n = 0}^{\infty }\mathcal{C} _{n, F}\left( p, q\right) \frac{t^{n}}{F_{n}!}\right) \\ & = &\sum\limits_{n = 0}^{\infty }\left( \sum\limits_{k = 0}^{n}\dbinom{n}{k}_{F}\mathcal{B} _{k, F}\left( \lambda \right) \mathcal{C}_{n-k}\left( p, q\right) \right) \frac{t^{n}}{F_{n}!}. \end{eqnarray*}$

Comparing the coefficients of $t^{n}$ on both sides of this last equation, we arrive at the desired result (3.20). Equation (3.21) can be similarly derived.

Theorem 3.4. The following identities hold true:

$\begin{equation} \mathcal{B}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left( -1\right) ^{k}q^{2k}\dbinom{n}{2k}_{F}\mathcal{B}_{n-2k, F}\left( p;\lambda \right) , \end{equation}$

(3.22)

and

$\begin{equation} \mathcal{B}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{\left\lfloor \frac{n-1}{2}\right\rfloor }\left( -1\right) ^{k}q^{2k+1}\dbinom{n}{2k+1}_{F}\mathcal{B}_{n-2k-1, F}\left( p;\lambda \right) . \end{equation}$

(3.23)

Proof. Setting $\alpha = 1$ in (3.5) and using (2.9), we find that

$\begin{eqnarray*} \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!} & = &\frac{t}{\lambda e_{F}^{t}-1}e_{F}^{pt}\cos _{F}\left( qt\right) \\ & = &\left( \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}\left( p;\lambda \right) \frac{t^{n}}{F_{n}!}\right) \left( \sum\limits_{n = 0}^{\infty }\left( -1\right) ^{n}q^{2n}\frac{t^{2n}}{F_{2n}!}\right) \\ & = &\sum\limits_{n = 0}^{\infty }\left( \sum\limits_{k = 0}^{\left\lfloor \frac{n}{2} \right\rfloor }\left( -1\right) ^{k}q^{2k}\dbinom{n}{2k}_{F}\mathcal{B} _{n-2k, F}\left( p;\lambda \right) \right) \frac{t^{n}}{F_{n}!}. \end{eqnarray*}$

Comparing the coefficients of $t^{n}$ on both sides of this last equation, we arrive at the desired result (3.22). Equation (3.23) can be similarly derived.

Theorem 3.5. The following identities hold true:

$\begin{equation} \mathcal{C}_{n, F}\left( p, q\right) = \lambda \sum\limits_{k = 0}^{n}\frac{1}{F_{k+1}} \dbinom{n}{k}_{F}\mathcal{B}_{n-k, F}^{(c, 1)}\left( p, q;\lambda \right) + \frac{\left( \lambda -1\right) }{F_{n+1}}\mathcal{B}_{n+1, F}^{(c, 1)}\left( p, q;\lambda \right) , \end{equation}$

(3.24)

and

$\begin{equation} \mathcal{S}_{n, F}\left( p, q\right) = \lambda \sum\limits_{k = 0}^{n}\frac{1}{F_{k+1}} \dbinom{n}{k}_{F}\mathcal{B}_{n-k, F}^{(s, 1)}\left( p, q;\lambda \right) + \frac{\left( \lambda -1\right) }{F_{n+1}}\mathcal{B}_{n+1, F}^{(s, 1)}\left( p, q;\lambda \right) . \end{equation}$

(3.25)

Proof. Using the following equation for the proof of (3.24), we have

$\begin{eqnarray*} e_{F}^{pt}\cos _{F}\left( qt\right) & = &\frac{\lambda e_{F}^{t}-1}{t}\frac{t}{ \lambda e_{F}^{t}-1}e_{F}^{pt}\cos _{F}\left( qt\right) \\ \sum\limits_{n = 0}^{\infty }\mathcal{C}_{n, F}\left( p, q\right) \frac{t^{n}}{F_{n}!} & = &\left( \lambda \sum\limits_{n = 0}^{\infty }\frac{t^{n-1}}{F_{n}!}-\frac{1}{t} \right) \left( \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!}\right) \\ & = &\left( \lambda \sum\limits_{n = 1}^{\infty }\frac{t^{n-1}}{F_{n}!}+\frac{\lambda -1 }{t}\right) \left( \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!}\right) . \end{eqnarray*}$

Considering $\mathcal{B}_{0, F}^{(c, 1)}\left(p, q; \lambda \right) = 0,$ and doing some calculations, we arrive at the desired result (3.24). Equation (3.25) can be similarly derived.

Theorem 3.6. The following identities hold true:

$\begin{equation} \mathcal{B}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}p^{k} \dbinom{n}{k}_{F}\mathcal{B}_{n-k, F}^{(c, 1)}\left( q;\lambda \right), \end{equation}$

(3.26)

and

$\begin{equation} \mathcal{B}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}p^{k} \dbinom{n}{k}_{F}\mathcal{B}_{n-k, F}^{(s, 1)}\left( q;\lambda \right). \end{equation}$

(3.27)

Proof. By applying (3.5), we have

$\begin{eqnarray*} \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) \frac{t^{n}}{F_{n}!} & = &\frac{t}{\lambda e_{F}^{t}-1}e_{F}^{pt}\cos _{F}\left( qt\right) \\ & = &\left( \sum\limits_{n = 0}^{\infty }\mathcal{B}_{n, F}^{(c, 1)}\left( q;\lambda \right) \frac{t^{n}}{F_{n}!}\right) \left( \sum\limits_{n = 0}^{\infty }p^{n}\frac{ t^{n}}{F_{n}!}\right) \\ & = &\sum\limits_{n = 0}^{\infty }\left( \sum\limits_{k = 0}^{n}p^{k}\dbinom{n}{k}_{F}\mathcal{B} _{n-k, F}^{(c, 1)}\left( q;\lambda \right) \right) \frac{t^{n}}{F_{n}!}. \end{eqnarray*}$

Comparing the coefficients of $t^{n}$ on both sides of this last equation, we arrive at the desired result (3.26). Equation (3.27) can be similarly derived.

Theorem 3.7. Determinantal forms of the cosine and sine Apostol Bernoulli-Fibonacci polynomials are given by

$\begin{equation*} \mathcal{B}_{n+1, F}^{(c, 1)}\left( p, q;\lambda \right) = \frac{1}{\left( \lambda -1\right) ^{n+2}}\left\vert \begin{array}{ccccc} F_{n+1}\mathcal{C}_{n, F}\left( p, q\right) & \lambda F_{n+1} & \lambda F_{n+1} \tbinom{n}{1}_{F} & \cdots & \lambda \tbinom{n}{n}_{F} \\ F_{n}\mathcal{C}_{n-1, F}\left( p, q\right) & \lambda -1 & \lambda F_{n} \tbinom{n-1}{0}_{F} & \cdots & \lambda \tbinom{n-1}{n-1}_{F} \\ F_{n-1}\mathcal{C}_{n-2, F}\left( p, q\right) & 0 & \lambda -1 & \cdots & \lambda \tbinom{n-2}{n-2}_{F} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ F_{0}\mathcal{C}_{-1, F}\left( p, q\right) & 0 & 0 & \cdots & \lambda -1 \end{array} \right\vert , \end{equation*}$

and

$\begin{equation*} \mathcal{B}_{n+1, F}^{(s, 1)}\left( p, q;\lambda \right) = \frac{1}{\left( \lambda -1\right) ^{n+2}}\left\vert \begin{array}{ccccc} F_{n+1}\mathcal{S}_{n, F}\left( p, q\right) & \lambda F_{n+1} & \lambda F_{n+1} \tbinom{n}{1}_{F} & \cdots & \lambda \tbinom{n}{n}_{F} \\ F_{n}\mathcal{S}_{n-1, F}\left( p, q\right) & \lambda -1 & \lambda F_{n} \tbinom{n-1}{0}_{F} & \cdots & \lambda \tbinom{n-1}{n-1}_{F} \\ F_{n-1}\mathcal{S}_{n-2, F}\left( p, q\right) & 0 & \lambda -1 & \cdots & \lambda \tbinom{n-2}{n-2}_{F} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ F_{0}\mathcal{S}_{-1, F}\left( p, q\right) & 0 & 0 & \cdots & \lambda -1 \end{array} \right\vert . \end{equation*}$

Proof. Equation (3.24) cause the system of unknown $\left(n+2\right)$ -equations with $\mathcal{B}_{n, F}^{(c, 1)}\left(p, q; \lambda \right),$ $\left(n = 0, 1, 2, \ldots \right).$ Then we apply the Cramer's rule to solve this equation. We obtain the desired result. In a similar way, we can obtain the determinantal form for sine Apostol Bernoulli-Fibonacci polynomials.

In subsections 3.2 and 3.3, we give the some basic properties of the polynomials $\mathcal{E}_{n, F}^{(c, \alpha)}\left(p, q; \lambda \right),$ $\mathcal{E}_{n, F}^{(s, \alpha)}\left(p, q; \lambda \right),$ $\mathcal{G} _{n, F}^{(c, \alpha)}\left(p, q; \lambda \right),$ and $\mathcal{G} _{n, F}^{(s, \alpha)}\left(p, q; \lambda \right).$ Their proofs run parallel to those of the results presented in this subsection; so, the proofs are omitted.

3.2. Basic properties of $\mathcal{E}_{n, F}^{(c, \alpha)}\left(p, q; \lambda \right)$ and $\mathcal{E}_{n, F}^{(s, \alpha)}\left(p, q; \lambda \right)$

Theorem 3.8. The following identities hold:

$\begin{equation*} \mathcal{E}_{n, F}^{(c, \alpha )}\left( p+r, q;\lambda \right) = \sum\limits_{k = 0}^{n}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{ E }_{k, F}^{(c, \alpha )}\left( p, q;\lambda \right) r^{n-k} \end{equation*}$

and

$\begin{equation*} \mathcal{E}_{n, F}^{(s, \alpha )}\left( p+r, q;\lambda \right) = \sum\limits_{k = 0}^{n}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{ E }_{k, F}^{(s, \alpha )}\left( p, q;\lambda \right) r^{n-k}. \end{equation*}$

Remark 3.6. We claim that

$\begin{equation*} \mathcal{E}_{n, F}^{(c, \alpha )}\left( p+1, q;\lambda \right) -\mathcal{E} _{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n-1}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{E}_{k, F}^{(c, \alpha )}\left( p, q;\lambda \right) , \end{equation*}$

$\begin{equation*} \mathcal{E}_{n, F}^{(s, \alpha )}\left( p+1, q;\lambda \right) -\mathcal{E} _{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n-1}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{E}_{k, F}^{(s, \alpha )}\left( p, q;\lambda \right) . \end{equation*}$

Theorem 3.9. For every $n\in \mathbb{N},$ following identities hold true:

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}p}\left\{ \mathcal{E}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \right\} = F_{n}\mathcal{E}_{n-1, F}^{(c, \alpha )}\left( p, q;\lambda \right) , \end{equation*}$

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}p}\left\{ \mathcal{E}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) \right\} = F_{n}\mathcal{E}_{n-1, F}^{(s, \alpha )}\left( p, q;\lambda \right) , \end{equation*}$

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}q}\left\{ \mathcal{E}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \right\} = -F_{n}\mathcal{E}_{n-1, F}^{(s, \alpha )}\left( p, q;\lambda \right) , \end{equation*}$

and

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}q}\left\{ \mathcal{E}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) \right\} = F_{n}\mathcal{E}_{n-1, F}^{(c, \alpha )}\left( p, q;\lambda \right) . \end{equation*}$

Theorem 3.10. The following identities hold true:

$\begin{equation*} \mathcal{E}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}\dbinom{n }{k}_{F}\mathcal{E}_{k, F}\left( \lambda \right) \mathcal{C}_{n-k}\left( p, q\right) , \end{equation*}$

and

$\begin{equation*} \mathcal{E}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}\dbinom{n }{k}_{F}\mathcal{E}_{k, F}\left( \lambda \right) \mathcal{S}_{n-k}\left( p, q\right) . \end{equation*}$

Theorem 3.11. The following identities hold true:

$\begin{equation*} \mathcal{E}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left( -1\right) ^{k}q^{2k}\dbinom{n}{2k}_{F}\mathcal{E}_{n-2k, F}\left( p;\lambda \right) , \end{equation*}$

and

$\begin{equation*} \mathcal{E}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{\left\lfloor \frac{n-1}{2}\right\rfloor }\left( -1\right) ^{k}q^{2k+1}\dbinom{n}{2k+1}_{F}\mathcal{E}_{n-2k-1, F}\left( p;\lambda \right) . \end{equation*}$

Theorem 3.12. The following identities hold true:

$\begin{equation*} \mathcal{C}_{n, F}\left( p, q\right) = \frac{1}{2}\mathcal{E} _{n, F}^{(c, 1)}\left( p, q;\lambda \right) +\frac{\lambda }{2}\sum\limits_{k = 0}^{n} \dbinom{n}{k}_{F}\mathcal{E}_{n-k, F}^{(c, 1)}\left( p, q;\lambda \right) , \end{equation*}$

and

$\begin{equation*} \mathcal{S}_{n, F}\left( p, q\right) = \frac{1}{2}\mathcal{E} _{n, F}^{(s, 1)}\left( p, q;\lambda \right) +\frac{\lambda }{2}\sum\limits_{k = 0}^{n} \dbinom{n}{k}_{F}\mathcal{E}_{n-k, F}^{(s, 1)}\left( p, q;\lambda \right) . \end{equation*}$

Theorem 3.13. The following identities hold true:

$\begin{equation*} \mathcal{E}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}p^{k} \dbinom{n}{k}_{F}\mathcal{E}_{n-k, F}^{(c, 1)}\left( q;\lambda \right) , \end{equation*}$

and

$\begin{equation*} \mathcal{E}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}p^{k} \dbinom{n}{k}_{F}\mathcal{E}_{n-k, F}^{(s, 1)}\left( q;\lambda \right) . \end{equation*}$

Theorem 3.14. Determinantal forms of the cosine and sine Apostol Euler-Fibonacci polynomials are given by

$\begin{equation*} \mathcal{E}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \left( \frac{2}{\lambda +1}\right) ^{n+1}\left\vert \begin{array}{ccccc} \mathcal{C}_{n, F}\left( p, q\right) & \frac{\lambda }{2}\tbinom{n}{1}_{F} & \frac{\lambda }{2}\tbinom{n}{2}_{F} & \cdots & \frac{\lambda }{2}\tbinom{n}{ n}_{F} \\ \mathcal{C}_{n-1, F}\left( p, q\right) & \frac{\lambda +1}{2} & \frac{\lambda }{2}\tbinom{n-1}{1}_{F} & \cdots & \frac{\lambda }{2}\tbinom{n-1}{n-1}_{F} \\ \mathcal{C}_{n-2, F}\left( p, q\right) & 0 & \frac{\lambda +1}{2} & \cdots & \frac{\lambda }{2}\tbinom{n-2}{n-2}_{F} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \mathcal{C}_{0, F}\left( p, q\right) & 0 & 0 & \cdots & \frac{\lambda +1}{2} \end{array} \right\vert , \end{equation*}$

and

$\begin{equation*} \mathcal{E}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \left( \frac{2}{\lambda +1}\right) ^{n+1}\left\vert \begin{array}{ccccc} \mathcal{S}_{n, F}\left( p, q\right) & \frac{\lambda }{2}\tbinom{n}{1}_{F} & \frac{\lambda }{2}\tbinom{n}{2}_{F} & \cdots & \frac{\lambda }{2}\tbinom{n}{ n}_{F} \\ \mathcal{S}_{n-1, F}\left( p, q\right) & \frac{\lambda +1}{2} & \frac{\lambda }{2}\tbinom{n-1}{1}_{F} & \cdots & \frac{\lambda }{2}\tbinom{n-1}{n-1}_{F} \\ \mathcal{S}_{n-2, F}\left( p, q\right) & 0 & \frac{\lambda +1}{2} & \cdots & \frac{\lambda }{2}\tbinom{n-2}{n-2}_{F} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \mathcal{S}_{0, F}\left( p, q\right) & 0 & 0 & \cdots & \frac{\lambda +1}{2} \end{array} \right\vert . \end{equation*}$

3.3. Basic properties of $\mathcal{G}_{n, F}^{(c, \alpha)}\left(p, q; \lambda \right)$ and $\mathcal{G}_{n, F}^{(s, \alpha)}\left(p, q; \lambda \right)$

Theorem 3.15. The following identities hold true:

$\begin{equation*} \mathcal{G}_{n, F}^{(c, \alpha )}\left( p+r, q;\lambda \right) = \sum\limits_{k = 0}^{n}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{ G }_{k, F}^{(c, \alpha )}\left( p, q;\lambda \right) r^{n-k}, \end{equation*}$

and

$\begin{equation*} \mathcal{G}_{n, F}^{(s, \alpha )}\left( p+r, q;\lambda \right) = \sum\limits_{k = 0}^{n}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{ G }_{k, F}^{(s, \alpha )}\left( p, q;\lambda \right) r^{n-k}. \end{equation*}$

Remark 3.7. We claim that

$\begin{equation*} \mathcal{G}_{n, F}^{(c, \alpha )}\left( p+1, q;\lambda \right) -\mathcal{G} _{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n-1}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{G}_{k, F}^{(c, \alpha )}\left( p, q;\lambda \right) , \end{equation*}$

$\begin{equation*} \mathcal{G}_{n, F}^{(s, \alpha )}\left( p+1, q;\lambda \right) -\mathcal{G} _{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n-1}\left( -1\right) ^{\tbinom{n-k}{2}}\dbinom{n}{k}_{F}\mathcal{G}_{k, F}^{(s, \alpha )}\left( p, q;\lambda \right) . \end{equation*}$

Theorem 3.16. For every $n\in \mathbb{N},$ following identities hold true:

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}p}\left\{ \mathcal{G}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \right\} = F_{n}\mathcal{G}_{n-1, F}^{(c, \alpha )}\left( p, q;\lambda \right) , \end{equation*}$

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}p}\left\{ \mathcal{G}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) \right\} = F_{n}\mathcal{G}_{n-1, F}^{(s, \alpha )}\left( p, q;\lambda \right) , \end{equation*}$

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}q}\left\{ \mathcal{G}_{n, F}^{(c, \alpha )}\left( p, q;\lambda \right) \right\} = -F_{n}\mathcal{G}_{n-1, F}^{(s, \alpha )}\left( p, q;\lambda \right) , \end{equation*}$

and

$\begin{equation*} \frac{\partial _{F}}{\partial _{F}q}\left\{ \mathcal{G}_{n, F}^{(s, \alpha )}\left( p, q;\lambda \right) \right\} = F_{n}\mathcal{G}_{n-1, F}^{(c, \alpha )}\left( p, q;\lambda \right) . \end{equation*}$

Theorem 3.17. The following identities hold true:

$\begin{equation*} \mathcal{G}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}\dbinom{n }{k}_{F}\mathcal{G}_{k}\left( \lambda \right) \mathcal{C}_{n-k}\left( p, q\right) , \end{equation*}$

and

$\begin{equation*} \mathcal{G}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}\dbinom{n }{k}_{F}\mathcal{G}_{k, F}\left( \lambda \right) \mathcal{S}_{n-k}\left( p, q\right) . \end{equation*}$

Theorem 3.18. The following identities hold true:

$\begin{equation*} \mathcal{G}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{\left\lfloor \frac{n}{2}\right\rfloor }\left( -1\right) ^{k}q^{2k}\dbinom{n}{2k}_{F}\mathcal{G}_{n-2k, F}\left( p;\lambda \right) , \end{equation*}$

and

$\begin{equation*} \mathcal{G}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{\left\lfloor \frac{n-1}{2}\right\rfloor }\left( -1\right) ^{k}q^{2k+1}\dbinom{n}{2k+1}_{F}\mathcal{G}_{n-2k-1, F}\left( p;\lambda \right) . \end{equation*}$

Theorem 3.19. The following identities hold true:

$\begin{equation*} \mathcal{C}_{n, F}\left( p, q\right) = \frac{\lambda }{2}\sum\limits_{k = 0}^{n}\frac{1}{ F_{k+1}}\dbinom{n}{k}_{F}\mathcal{G}_{n-k, F}^{(c, 1)}\left( p, q;\lambda \right) +\frac{\lambda +1}{2F_{n+1}}\mathcal{G}_{n+1, F}^{(c, 1)}\left( p, q;\lambda \right) , \end{equation*}$

and

$\begin{equation*} \mathcal{S}_{n, F}\left( p, q\right) = \frac{\lambda }{2}\sum\limits_{k = 0}^{n}\frac{1}{ F_{k+1}}\dbinom{n}{k}_{F}\mathcal{G}_{n-k, F}^{(s, 1)}\left( p, q;\lambda \right) +\frac{\lambda +1}{2F_{n+1}}\mathcal{G}_{n+1, F}^{(s, 1)}\left( p, q;\lambda \right) . \end{equation*}$

Theorem 3.20. The following identities hold true:

$\begin{equation*} \mathcal{G}_{n, F}^{(c, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}p^{k} \dbinom{n}{k}_{F}\mathcal{G}_{n-k, F}^{(c, 1)}\left( q;\lambda \right) , \end{equation*}$

and

$\begin{equation*} \mathcal{G}_{n, F}^{(s, 1)}\left( p, q;\lambda \right) = \sum\limits_{k = 0}^{n}p^{k} \dbinom{n}{k}_{F}\mathcal{G}_{n-k, F}^{(s, 1)}\left( q;\lambda \right) . \end{equation*}$

Theorem 3.21. Determinantal forms of the cosine and sine Apostol Genocchi-Fibonacci polynomials are given by

$\begin{equation*} \mathcal{G}_{n+1, F}^{(c, 1)}\left( p, q;\lambda \right) = \left( \frac{2}{ \lambda +1}\right) ^{n+2}\left\vert \begin{array}{ccccc} F_{n+1}\mathcal{C}_{n, F}\left( p, q\right) & \frac{\lambda }{2}F_{n+1} \tbinom{n}{0}_{F} & \frac{\lambda }{2}F_{n+1}\tbinom{n}{1}_{F} & \cdots & \frac{\lambda }{2}\tbinom{n}{n}_{F} \\ F_{n}\mathcal{C}_{n-1, F}\left( p, q\right) & \frac{\lambda +1}{2} & \frac{ \lambda }{2}F_{n}\tbinom{n-1}{0}_{F} & \cdots & \frac{\lambda }{2}\tbinom{ n-1}{n-1}_{F} \\ F_{n-1}\mathcal{C}_{n-2, F}\left( p, q\right) & 0 & \frac{\lambda +1}{2} & \cdots & \frac{\lambda }{2}\tbinom{n-2}{n-2}_{F} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \frac{\lambda +1}{2} \end{array} \right\vert , \end{equation*}$

and

$\begin{equation*} \mathcal{G}_{n+1, F}^{(s, 1)}\left( p, q;\lambda \right) = \left( \frac{2}{ \lambda +1}\right) ^{n+2}\left\vert \begin{array}{ccccc} F_{n+1}\mathcal{S}_{n, F}\left( p, q\right) & \frac{\lambda }{2}F_{n+1} \tbinom{n}{0}_{F} & \frac{\lambda }{2}F_{n+1}\tbinom{n}{1}_{F} & \cdots & \frac{\lambda }{2}\tbinom{n}{n}_{F} \\ F_{n}\mathcal{S}_{n-1, F}\left( p, q\right) & \frac{\lambda +1}{2} & \frac{ \lambda }{2}F_{n}\tbinom{n-1}{0}_{F} & \cdots & \frac{\lambda }{2}\tbinom{ n-1}{n-1}_{F} \\ F_{n-1}\mathcal{S}_{n-2, F}\left( p, q\right) & 0 & \frac{\lambda +1}{2} & \cdots & \frac{\lambda }{2}\tbinom{n-2}{n-2}_{F} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & \frac{\lambda +1}{2} \end{array} \right\vert . \end{equation*}$

4. Conclusions

Our aim in this article is to define the $F$ -analogues of the parametric types of the Apostol Bernoulli, the Apostol Euler, and the Apostol Genocchi polynomials studied by Srivastava et al. ^[15,19]. Namely, we have defined parametric types of the Apostol Bernoulli-Fibonacci, the Apostol Euler-Fibonacci, and the Apostol Genocchi-Fibonacci polynomials using the Golden calculus and investigated their properties. In our future work, we plan to define the parametric types of some special polynomials with the help of Golden calculus and to obtain many combinatorial identities with the help of their generating functions.

Conflict of interest

All authors declare no conflicts of interest in this paper.

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