Some fractional integral inequalities for the Katugampola integral operator

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]):

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]):

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]):

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

and

and

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:

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

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:

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

in terms of the Golden binomial coefficients, called as Fibonomials

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

and

The Golden derivative defined as follows:

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

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

and

Briefly, we use the following notations throughout the paper

and

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

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

and

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

and

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:

and

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

then their fibonomial convolution is defined as

So, the generating function takes the form

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:

and

where

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:

whose exponential generating functions are given, respectively, by

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

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:

and

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

which readily yields

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

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

Remark 3.5. We claim that

and

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

and

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

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

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:

and

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

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:

and

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

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:

and

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

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:

and

Proof. By applying (3.5), we have

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

and

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:

and

Remark 3.6. We claim that

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

and

Theorem 3.10. The following identities hold true:

and

Theorem 3.11. The following identities hold true:

and

Theorem 3.12. The following identities hold true:

and

Theorem 3.13. The following identities hold true:

and

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

and

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:

and

Remark 3.7. We claim that

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

and

Theorem 3.17. The following identities hold true:

and

Theorem 3.18. The following identities hold true:

and

Theorem 3.19. The following identities hold true:

and

Theorem 3.20. The following identities hold true:

and

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

and

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.