1 Introduction

Recently, many specialists have investigated special functions due to the important role of these functions in mathematical, physical and engineering problems. Various extensions of some special functions were studied in many works (see, e.g., [3–6]).

For introducing some new weighted hypergeometric functions, we use the weighted extension of Euler’s beta-function (a particular case associated with the extensions considered in [7]):

$$ B_{\omega}^{(\alpha,\beta)}(x,y)= \int_{0}^{1} t^{x-1}(1-t)^{y-1} \omega ^{(\alpha,\beta)}(zt,u,v)\,dt, $$
(1.1)

where \(\operatorname{Re}x>0\), \(\operatorname{Re}y>0\) and α, β, z, u, v are real or complex parameters and \(\omega^{(\alpha,\beta)}(zt,u,v)\) is a function of the class Ω, i.e. such that the integral (1.1) is absolutely convergent. Besides, by writing \(B_{\omega}(x,y)\) we mean that this function and the function ω do not depend on α and β. One can see that, if \(\omega (t,p,0)=e^{\frac{-p}{t(1-t)}}\) with \(\operatorname{Re}p>0\), then \(B_{\omega }(x,y)\) (\(\min \{\operatorname{Re}x, \operatorname{Re}y\}>0\)) becomes the extension of Euler’s beta-function considered by Chaudhry et al. [8]. Besides, by a straightforward calculation, it follows in the general case that

$$B_{\omega}(x,y+1)+B_{\omega}(x+1,y)=B_{\omega}(x,y),\quad \omega\in\Omega. $$

In the last years, the interest to application of the fractional derivative operators a considerable growth. Numerous publications have been devoted to the solutions of different problems by applying these operators (see [9–11]).

Recall that the well-known Caputo fractional derivative is defined as

$$D_{z}^{\mu}f(z):=\frac{1}{\Gamma(m-\mu)} \int_{0}^{z} (z-t)^{m-\mu-1} \frac {d^{m}}{dt^{m}}f(t)\,dt, $$

where \(m-1<\operatorname{Re}\mu<m\), \(m\in\mathbb{N}\) (the set of positive integers). Here and in the following, let \(\mathbb{C}\), \(\mathbb{R}\), \(\mathbb{N}\), and \({\mathbb {Z}}_{0}^{-}\) be the sets of complex numbers, real numbers, positive integers, and non-positive integers, respectively. We introduce the general Caputo fractional derivative suitable with the function (1.1) by the following generalization of that defined in [12, 13]:

$$ D_{z}^{\mu,\tau}f(z):=\frac{1}{\Gamma(m-\mu)} \int_{0}^{z} (z-t)^{m-\mu-1}\tau (t,u,v) \frac{d^{m}}{dt^{m}}f(t)\,dt, $$
(1.2)

where \(m-1<\operatorname{Re}\mu<m\), \(m\in\mathbb{N}\) and u, v are real or complex parameters and τ is assumed to be a function of the class Λ, i.e. such that the integral (1.2) is absolutely convergent. It is noted that (1.2) becomes the extended Caputo fractional derivative [12] when \(\tau (t,p,z)=e^{\frac{-pz^{2}}{t(z-t)}}\) (\(\operatorname{Re}p>0\)), while it becomes the classical Caputo fractional derivative when \(\tau\equiv1\).

2 Weighted hypergeometric functions

In the same form as defined in [12], we introduce some weighted versions of the Gauss hypergeometric function \(_{2}{F_{1}}\), the Appell hypergeometric functions \(F_{1}\), \(F_{2}\) (see [14]), the Lauricella hypergeometric function \(F^{3}_{D;\omega}\), the generalized Gauss hypergeometric function \(F_{\omega}\) and the generalized confluent hypergeometric function \({_{1}}F_{1}^{\omega}\). Everywhere in this paper we assume that \(m\in\mathbb{N}\), \(\omega\in\Omega\) and \(B(x,y)\) (\(\min\{\operatorname{Re}x, \operatorname{Re}y\}>0\)) is the classical Beta function.

Definition 2.1

The ω-weighted extended Gauss hypergeometric function is

$$ _{2}{F_{1}}(a,b;c;z;\omega):=\sum _{n=0}^{\infty}\frac {(a)_{n}(b)_{n}}{(b-m)_{n}}\frac{B_{\omega}(b-m+n,c-b+m)}{B(b-m,c-b+m)} \frac{z^{n}}{n!}, $$
(2.1)

where \(\vert z \vert <1\) and \(m<\operatorname{Re}b<\operatorname{Re}c\). For an illustration of the Gauss hypergeometric function see [15, 16].

Definition 2.2

The ω-weighted extended Appell hypergeometric function \(F_{1}\) is

$$ F_{1}(a,b,c;d;x,y;\omega):=\sum _{n,k=0}^{\infty}\frac {(a)_{n+k}(b)_{n}(c)_{k}}{(a-m)_{n+k}}\frac{B_{\omega}(a-m+n+k,d-a+m)}{B(a-m,d-a+m)} \frac{x^{n}}{n!}\frac{y^{k}}{k!}, $$
(2.2)

where \(\vert x \vert <1\), \(\vert y \vert <1\) and \(m<\operatorname{Re}a<\operatorname{Re}d\).

Definition 2.3

The ω-weighted extended Appell hypergeometric function \(F_{2}\) is

$$ \begin{aligned}[b] &F_{2}(a,b,c;d,e;x,y;\omega) \\ &\quad := \sum_{n,k=0}^{\infty}\frac{(a)_{n+k}(b)_{n}(c)_{k}}{(b-m)_{n}(c-m)_{k}} \frac {B_{\omega}(b-m+n,d-b+m)}{B(b-m,d-b+m)} \\ &\quad\quad{} \times \frac{B_{\omega}(c-m+k,e-c+m)}{B(c-m,e-c+m)}\frac{x^{n}}{n!}\frac{y^{k}}{k!}, \end{aligned} $$
(2.3)

where \(\vert x \vert + \vert y \vert <1\), \(m< \operatorname{Re}b<\operatorname{Re}d\) and \(m<\operatorname{Re}c<\operatorname{Re}e\). Besides, we just consider one of the form of Appell hypergeometric function [12, 17].

Definition 2.4

The ω-weighted extended Appell hypergeometric function \(F_{D,\omega}^{3}\) is

$$ \begin{aligned}[b] &F_{D,\omega}^{3}(a,b,c,d;e;x,y,z;\omega) \\ &\quad := \sum_{n,k,r=0}^{\infty}\frac {(a)_{n+k+r}(b)_{n}(c)_{k}(d)_{r}}{(a-m)_{n+k+r}} \frac{B_{\omega}(a-m+n+k+r,e-a+m)}{B(a-m,e-a+m)}\frac{x^{n}}{n!}\frac{y^{k}}{k!}\frac {z^{r}}{r!}, \end{aligned} $$
(2.4)

where \(\sqrt{ \vert x \vert }+\sqrt{ \vert y \vert }+\sqrt{ \vert z \vert }<1\) and \(m<\operatorname{Re}a<\operatorname{Re}e\).

Note that the functions defined above become those in [12] when \(\omega(t,p,0)=e^{\frac{-p}{t(1-t)}}\) and \(\operatorname{Re}p>0\). Besides, for \(\omega\equiv1\), these functions reduce to the well-known Gauss hypergeometric function \(_{2}{F_{1}}\), Appell functions \(F_{1}\), \(F_{2}\) and Lauricella function \(F^{3}_{D}\), respectively.

Definition 2.5

The ω-weighted generalized Gauss hypergeometric function \(F_{\omega}\) is

$$ F_{\omega}(a,b;c;z)=\sum_{n=0}^{\infty}(a)_{n} \frac{B_{\omega}(b+n,c-b)}{B(b,c-b)}\frac{z^{n}}{n!}, $$
(2.5)

where \(\vert z \vert <1\) and \(\operatorname{Re}c>\operatorname{Re}b>0\).

Definition 2.6

The ω-weighted generalized confluent hypergeometric function \({_{1}}F_{1}^{\omega}\) is

$$ {} _{1}F_{1}^{\omega}(b;c;z)=\sum _{n=0}^{\infty}\frac{B_{\omega}(b+n,c-b)}{B(b,c-b)}\frac{z^{n}}{n!}, $$
(2.6)

where \(\vert z \vert <1\) and \(\operatorname{Re}c>\operatorname{Re}b>0\).

For some examples of functions \(F_{\omega}\) and \(_{1}F_{1}^{\omega}\), see [7].

Remark 2.1

If \(\omega^{(\alpha,\beta)}(t,p,0)={{}_{1} F_{1} (\alpha;\beta;\frac {-p}{t(1-t)} )}\) where \(\min\{\operatorname{Re}x, \operatorname{Re}y, \operatorname{Re}\alpha, \operatorname{Re}\beta\}>0\) and \(\operatorname{Re}p\geq0\), then \(B^{(\alpha,\beta)}_{\omega}(x,y)\) is the function \(B_{p}^{(\alpha ,\beta)}(x,y)\) defined in [5] (see p.32 in [5], also, p.1748 in [18]). Hence, \(F_{\omega}(a,b;c;z)=F_{p}^{(\alpha,\beta )}(a,b;c;z)\) and \(_{1} F_{1}^{\omega}(b;c;z)={_{1}F_{1}^{(\alpha,\beta ,p)}(b;c;z)}\) are the same as those in [18], pp.1748-1749 (see also [5], p.39).

The next definition, it will be useful to introduce a general result.

Definition 2.7

Let \(f(z):= \sum_{n=0}^{\infty}a_{n} z^{n}\) and \(g(z):= \sum_{n=0}^{\infty}b_{n} z^{n}\) be two power series whose convergence or radii are \(R_{f}\) and \(R_{g}\), respectively. Then the Hadamard product of \(f(z)\) and \(g(z)\) is the power series (see [18, 19])

$$(f*g) (z):=\sum_{n=0}^{\infty}a_{n} b_{n} z^{n}, $$

whose convergence radius R satisfies the inequality \(R_{f}R_{g}\leq R\).

Remark 2.2

The above definitions can be considered with \(B_{\omega}^{(\alpha,\beta )}\) instead of \(B_{\omega}\).

3 The weighted Caputo derivative

Below, we establish some useful statements, where we apply the weighted Caputo fractional derivative. The next results are some generalizations of those in [12, 14, 18] and some others. Hence, the tools to prove them (lemmas and theorems) are similar.

Lemma 3.1

If \(m-1<\operatorname{Re}\mu<m\), \(\omega\in\Lambda\), \(\omega\in\Omega\), \(s,w\in\mathbb{C}\) and \(\operatorname{Re}\mu<\operatorname{Re}\lambda\), then

$$D_{z}^{\mu,\omega} \bigl[z^{\lambda} \bigr]= \frac{\Gamma(\lambda+1)B_{\omega}(\lambda -m+1,m-\mu)}{\Gamma(\lambda-\mu+1)B(\lambda-m+1,m-\mu)}z^{\lambda-\mu}. $$

Proof

Indeed,

$$\begin{aligned} D_{z}^{\mu,\omega} \bigl[z^{\lambda} \bigr]&=\frac{1}{\Gamma(m-\mu)} \int_{0}^{z} (z-t)^{m-\mu-1}\omega(t,s,w) \frac{d^{m}}{dt^{m}}t^{\lambda}\,dt \\ &=\frac{\Gamma(\lambda+1)}{\Gamma(m-\mu)\Gamma(\lambda-m+1)} \int_{0}^{z} (z-t)^{m-\mu-1}t^{\lambda-m} \omega(t,s,w)\,dt \\ &=\frac{\Gamma(\lambda+1)z^{\lambda-\mu}}{\Gamma(m-\mu)\Gamma(\lambda -m+1)} \int_{0}^{1} (1-u)^{m-\mu-1}u^{\lambda-m} \omega(zu,s,w)\,du \\ &=\frac{\Gamma(\lambda+1)B_{\omega}(\lambda-m+1,m-\mu)}{\Gamma(\lambda-\mu +1)B(\lambda-m+1,m-\mu)}z^{\lambda-\mu}. \end{aligned}$$

 □

Theorem 3.1

If \(f(z)=\sum_{n=0}^{\infty}a_{n} z^{n}\) is an analytic function in the disk \(\vert z \vert <\rho\), \(\omega\in\Lambda \) and \(\omega\in\Omega\), then

$$D_{z}^{\mu,\omega} \bigl\{ f(z) \bigr\} =\sum _{n=0}^{\infty}D_{z}^{\mu,\omega} \bigl\{ z^{n} \bigr\} , $$

where \(m-1<\operatorname{Re}\mu<m\).

Proof

As the power series converges uniformly, the integral of \(D_{z}^{\mu ,\omega}\{f(z)\}\) converges absolutely, and hence the desired result follows by a straightforward calculation. □

Theorem 3.2

Let \(m-1<\operatorname{Re}\lambda-\mu<m<\operatorname{Re}\lambda\), \(\kappa\in \mathbb{C}\), \(\omega\in\Lambda\) and \(\omega\in\Omega\). Then

$$\begin{aligned} D_{z}^{\lambda-\mu,\omega} \bigl\{ z^{\lambda-1}(1-z)^{-\kappa} \bigr\} =z^{\mu-1}\frac{\Gamma(\lambda)}{\Gamma(\mu)}{_{2}} {F_{1}( \kappa,\lambda;\mu ;z;\omega)},\quad \vert z \vert < 1. \end{aligned}$$
(3.1)

Proof

Using the power series of \((1-z)^{-\kappa}\), Theorem 3.1, Lemma 3.1 and equation (2.1), we obtain

$$\begin{aligned} D_{z}^{\lambda-\mu,\omega} \bigl\{ z^{\lambda-1}(1-z)^{-\kappa} \bigr\} &=D_{z}^{\lambda -\mu,\omega} \Biggl\{ z^{\lambda-1}\sum _{n=0}^{\infty}(\kappa)_{n}\frac {z^{n}}{n!} \Biggr\} \\ &=\sum_{n=0}^{\infty}\frac{(\kappa)_{n}}{n!}D_{z}^{\lambda -\mu,\omega} \bigl\{ z^{\lambda+n-1} \bigr\} \\ &=z^{\mu-1}\sum_{n=0}^{\infty} \frac{(\kappa)_{n} \Gamma(\lambda+n)B_{\omega}(\lambda-m+n,m-\lambda+\mu)}{\Gamma(\mu+n)B(\lambda-m+n,m-\lambda+\mu )}\frac{z^{n}}{n!} \\ &=z^{\mu-1}\frac{\Gamma(\lambda)}{\Gamma(\mu)}\sum_{n=0}^{\infty} \frac {(\kappa)_{n}(\lambda)_{n}}{(\lambda-m)_{n}}\frac{B_{\omega}(\lambda -m+n,m-\lambda+\mu)}{B(\lambda-m,\mu-\lambda+m)}\frac{z^{n}}{n!} \\ &=z^{\mu-1}\frac{\Gamma(\lambda)}{\Gamma(\mu)}{_{2}} {F_{1}(\kappa, \lambda ;\mu;z,\omega)}. \end{aligned}$$

 □

Theorem 3.3

If \(m-1<\operatorname{Re}(\lambda-\mu)<m<\operatorname{Re}\lambda\), \(\omega\in \Lambda\) and \(\omega\in\Omega\), then

$$\begin{aligned} &D_{z}^{\lambda-\mu,\omega} \bigl\{ z^{\lambda-1}(1-rz)^{-\kappa }(1-sz)^{-\theta} \bigr\} =z^{\mu-1}\frac{\Gamma(\lambda)}{\Gamma(\mu )}F_{1}(\lambda,\kappa,\theta; \mu;rz;sz;\omega), \end{aligned}$$
(3.2)

for \(r,s,\kappa,\theta\in\mathbb{C}\), \(\vert rz \vert <1\) and \(\vert sz \vert <1\).

Proof

Using the power series of \((1-rz)^{-\kappa}\), \((1-sz)^{-\theta}\), Theorem 3.1, Lemma 3.1 and (2.2), we get

$$\begin{aligned} &D_{z}^{\lambda-\mu,\omega} \bigl\{ z^{\lambda-1}(1-rz)^{-\kappa}(1-sz)^{-\theta } \bigr\} \\ &\quad =D_{z}^{\lambda-\mu,\omega} \Biggl(\sum_{n,k=0}^{\infty} \frac{(\kappa)_{n} (\theta)_{k} r^{n} s^{k} z^{\lambda+n+k-1}}{n!k!} \Biggr) \\ &\quad =\sum_{n,k=0}^{\infty}\frac{(\kappa)_{n} (\theta)_{k} r^{n} s^{k} }{n!k!} D_{z}^{\lambda-\mu,\omega} \bigl\{ z^{\lambda+n+k-1} \bigr\} \\ &\quad =z^{\mu-1}\sum_{n,k=0}^{\infty} \frac{(\kappa)_{n} (\theta)_{k} r^{n} s^{k} }{n!k!}\frac{\Gamma(\lambda+n+k)B_{\omega}(\lambda-m+n+k,m-\lambda+\mu )}{\Gamma(\mu+n+k)B(\lambda-m+n+k,m-\lambda+\mu)}z^{n+k} \\ &\quad =z^{\mu-1}\frac{\Gamma(\lambda)}{\Gamma(\mu)}\sum_{n,k=0}^{\infty} \frac {(\lambda)_{n+k}(\kappa)_{n}(\theta)_{k}}{(\lambda-m)_{n+k}}\frac{B_{\omega}(\lambda-m+n+k,m-\lambda+\mu)}{B(\lambda-m,m-\lambda+\mu)}\frac {(rz)^{n}}{n!}\frac{(sz)^{k}}{k!} \\ &\quad =z^{\mu-1}\frac{\Gamma(\lambda)}{\Gamma(\mu)}F_{1}(\lambda,\kappa,\theta ; \mu;rz;sz;\omega). \end{aligned}$$

 □

Theorem 3.4

If \(m-1<\operatorname{Re}(\lambda-\mu)<m<\operatorname{Re}\lambda\), \(\omega\in \Lambda\), \(\omega\in\Omega\) and \(m<\operatorname{Re}\beta<\operatorname{Re}\gamma\), then

$$ \begin{aligned}[b] &D_{z}^{\lambda-\mu,\omega} \biggl(z^{\lambda-1}(1-z)^{-\alpha}{_{2}{F_{1} \biggl(\alpha,\beta;\gamma;\frac{x}{1-z};\omega \biggr)}} \biggr) \\ &\quad=\frac{\Gamma(\lambda)}{\Gamma(\mu)}z^{\mu-1}F_{2}(\alpha,\beta ,\lambda, \gamma;\mu;x,z;\omega),\quad \vert x \vert + \vert z \vert < 1. \end{aligned} $$
(3.3)

Proof

By the power series of \((1-az)^{-\alpha}\), (2.1) and (3.1), we obtain

$$\begin{aligned} &D_{z}^{\lambda-\mu,\omega} \biggl(z^{\lambda-1}(1-z)^{-\alpha}{_{2}{F_{1} \biggl(\alpha,\beta;\gamma;\frac{x}{1-z};\omega \biggr)}} \biggr) \\ &\quad =D_{z}^{\lambda-\mu,\omega} \bigl\{ z^{\lambda-1}(1-z)^{-\alpha-n} \bigr\} \times\sum_{n=0}^{\infty} \frac{(\alpha)_{n}(\beta)_{n}}{(\beta-m)_{n}} \frac{B_{\omega}(\beta-m+n,\gamma-\beta+m)}{B(\beta-m,\gamma-\beta+m)}\frac{x^{n}}{n!} \\ &\quad =z^{\mu-1}\frac{\Gamma(\lambda)}{\Gamma(\mu)}\sum_{n,k=0}^{\infty} \frac {(\alpha)_{n+k}(\beta)_{n} (\lambda)_{k}}{(\beta-m)_{n} (\lambda-m)_{k}}\frac {B_{\omega}(\beta-m+n,\gamma-\beta+m)}{B(\beta-m,\gamma-\beta+m)} \\ &\quad\quad{}\times \frac {B_{\omega}(\lambda-m+k,\mu-\lambda+m)}{B(\lambda-m,\mu-\lambda+m)}\frac {x^{n}}{n!} \frac{z^{k}}{k!} \\ &\quad =z^{\mu-1}\frac{\Gamma(\lambda)}{\Gamma(\mu)}F_{2}(\lambda,\alpha,\beta ; \mu;az;bz;\omega). \end{aligned}$$

 □

4 Generating functions

Below we obtain some bilinear generating relations for the weighted extended hypergeometric function \(_{2}{F_{1}}\). In a sense, these relations are similar to those in [7, 12] by taking some particular weights ω belonging to \(\Lambda\cap\Omega\).

Theorem 4.1

If \(m-1<\operatorname{Re}(\lambda-\mu)<\operatorname{Re}\mu\), \(\omega\in \Lambda\) and \(\omega\in\Omega\), then

$$ \sum_{n=0}^{\infty} \frac{(\alpha)_{n}}{n!}{_{2}} {F_{1}(\alpha+n,\lambda;\mu ;z; \omega)}t^{n}=(1-t)^{-\alpha}{_{2}} {F_{1} \biggl(\alpha,\lambda;\mu;\frac {z}{1-t};\omega \biggr)}, $$
(4.1)

where \(\vert z \vert <\min \{1, \vert 1-t \vert \}\).

Proof

Note that by the identity of [12]

$$\bigl[(1-z)-t \bigr]^{-\alpha}=(1-t)^{-\alpha} \biggl(1- \frac{z}{1-t} \biggr)^{-\alpha }. $$

We write a power series in the left hand side

$$\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}(1-z)^{-\alpha} \biggl(\frac {t}{1-z} \biggr)^{n}=(1-t)^{-\alpha} \biggl(1- \frac{z}{1-t} \biggr)^{-\alpha }, $$

where \(\vert t \vert < \vert 1-z \vert \). Besides, by multiplying both sides by \(z^{\lambda-1}\) and applying our operator \(D_{z}^{\lambda-\mu,\omega}\), we get

$$D_{z}^{\lambda-\mu,\omega} \Biggl\{ \sum_{n=0}^{\infty} \frac{(\alpha)_{n} t^{n}}{n!}z^{\lambda-1}(1-z)^{-\alpha-n} \Biggr\} =D_{z}^{\lambda-\mu,\omega } \biggl\{ (1-t)^{-\alpha}z^{\lambda-1} \biggl(1-\frac{z}{1-t} \biggr)^{-\alpha} \biggr\} . $$

As \(\vert t \vert < \vert 1-z \vert \) and \(0<\operatorname{Re}\mu<\operatorname{Re}\lambda\), the fractional derivative can be replaced inside the sum:

$$\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}D_{z}^{\lambda-\mu,\omega} \bigl\{ z^{\lambda-1}(1-z)^{-\alpha-n} \bigr\} t^{n}=(1-t)^{-\alpha}D_{z}^{\lambda -\mu,\omega} \biggl\{ z^{\lambda-1} \biggl(1-\frac{z}{1-t} \biggr)^{-\alpha } \biggr\} . $$

Finally, by Theorem 3.2, we get (4.1). □

Theorem 4.2

If \(m-1<\operatorname{Re}(\lambda-\mu)<\operatorname{Re}\mu\), \(\omega\in \Lambda\) and \(\omega\in\Omega\), then

$$ \sum_{n=0}^{\infty} \frac{(\alpha)_{n}}{n!}{_{2}{F_{1}(\beta-n,\lambda;\mu ;z; \omega)}}t^{n}=(1-t)^{-\alpha}F_{1} \biggl(\beta, \alpha, \lambda;\mu ;z;\frac{zt}{1-t};\omega \biggr), $$
(4.2)

where \(\vert t \vert <\frac{1}{1+ \vert z \vert }\).

Proof

By the identity of [12] we get

$$\bigl[1-(1-z)t \bigr]^{-\alpha}=(1-t)^{-\alpha} \biggl(1+ \frac{-zt}{1-t} \biggr)^{-\alpha}, $$

and we write the power series in the left hand side

$$\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}(1-z)^{n} t^{n}=(1-t)^{-\alpha} \biggl(1-\frac{-zt}{1-t} \biggr)^{-\alpha},\quad \vert t \vert < 1/ \vert 1-z \vert . $$

Besides, by multiplying both sides by \(z^{\lambda-1}(1-z)^{-\beta}\) and applying the weighted Caputo fractional derivative \(D_{z}^{\lambda-\mu ,\omega}\) we get

$$D_{z}^{\lambda-\mu,\omega} \Biggl\{ \sum_{n=0}^{\infty} \frac{(\alpha )_{n}}{n!}z^{\lambda-1}(1-z)^{-\beta-n}t^{n} \Biggr\} =D_{z}^{\lambda-\mu ,\omega} \biggl\{ (1-t)^{-\alpha}z^{\lambda-1}(1-z)^{-\beta} \biggl(1-\frac {-zt}{1-t} \biggr)^{-\alpha} \biggr\} . $$

As \(\vert zt \vert < \vert 1-t \vert \) and \(0< \operatorname{Re}\mu<\operatorname{Re}\lambda\), the derivative can be replaced inside the sum:

$$\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}D_{z}^{\lambda-\mu,\omega} \bigl\{ z^{\lambda-1}(1-z)^{-\alpha-n} \bigr\} t^{n}=(1-t)^{-\alpha}D_{z}^{\lambda -\mu,\omega} \biggl\{ z^{\lambda-1}(1-z)^{-\beta} \biggl(1-\frac {-zt}{1-t} \biggr)^{-\alpha} \biggr\} . $$

Hence, equation (4.2) follows by Theorems 3.2 and 3.3. □

Theorem 4.3

If \(m-1<\operatorname{Re}(\beta-\gamma)<\operatorname{Re}\beta\), \(m<\operatorname{Re}\lambda<\operatorname{Re}\mu\), \(\omega\in\Lambda\) and \(\omega\in\Omega \), then

$$ \sum_{n=0}^{\infty} \frac{(\alpha)_{n}}{n!}{_{2}{F_{1}(\alpha+n,\lambda;\mu ;z; \omega)}} {_{2}{F_{1}(-n,\beta;\gamma;u;\omega)}}= F_{2} \biggl(\alpha,\lambda ,\beta;\mu,\gamma;z,\frac{ut}{1-t}; \omega \biggr). $$
(4.3)

Proof

If t tends to \((1-u)t\) in equation (4.1) and multiplying both sides by \(u^{\beta-1}\), we obtain

$$\begin{aligned} &\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}{_{2}} {F_{1}(\alpha+n,\lambda;\mu ;z;\omega)}u^{\beta-1}(1-u)^{n} t^{n} \\ &\quad =u^{\beta-1} \bigl[1-(1-u)t \bigr]^{-\alpha }{_{2}} {F_{1} \biggl(\alpha,\lambda;\mu;\frac{z}{1-(1-u)t};\omega \biggr)}. \end{aligned} $$

Hence, applying the fractional derivative \(D_{u}^{\beta-\lambda,\omega}\) to both sides we get

$$\begin{aligned} &\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}{_{2}} {F_{1}(\alpha+n,\lambda;\mu ;z;\omega)}D_{u}^{\beta-\lambda,\omega} \bigl\{ u^{\beta-1}(1-u)^{n} \bigr\} t^{n} \\ &\quad= D_{u}^{\beta-\lambda,\omega} \biggl\{ u^{\beta-1} \bigl[1-(1-u)t \bigr]^{-\alpha }{_{2}} {F_{1} \biggl(\alpha,\lambda;\mu; \frac{z}{1-(1-u)t};\omega \biggr)} \biggr\} , \end{aligned}$$

where \(\vert z \vert <1\), \(\vert \frac{1-u}{1-z}t \vert <1\) and \(\vert \frac{z}{1-t} \vert + \vert \frac {ut}{1-t} \vert <1\). This formula is the same as

$$\begin{aligned} &\sum_{n=0}^{\infty}\frac{(\alpha)_{n}}{n!}{_{2}} {F_{1}(\alpha+n,\lambda;\mu ;z;\omega)}D_{u}^{\beta-\lambda,\omega} \bigl\{ u^{\beta-1}(1-u)^{n} \bigr\} t^{n} \\ &\quad= D_{u}^{\beta-\lambda,\omega} \biggl\{ u^{\beta-1} \biggl[1- \frac {-ut}{1-t} \biggr]^{-\alpha}{_{2}} {F_{1} \biggl( \alpha,\lambda;\mu;\frac {z}{1-\frac{-ut}{1-t}};\omega \biggr)} \biggr\} . \end{aligned}$$

Therefore, we obtain the desired statement (4.3) by Theorems 3.2 and 3.4. □

Now, using the weighted function \(F_{\omega}\), we introduce a generalization of the generating relation given in [18], pp.1750-1751 (see also [14]).

Theorem 4.4

If \(\operatorname{Re}c>\operatorname{Re}b>0\) and \(\omega\in\Omega\), then

$$\begin{aligned} &(1+t)^{-\lambda}F_{\omega}\bigl(a,b;c,z/(1+t) \bigr) \\ &\quad =\sum_{r=0}^{\infty}(-1)^{r} ( \lambda)_{r} F_{\omega}(a,b;c;z)\ast{_{1} {F_{1}(\lambda+r;\lambda;z)}}\frac{t^{r}}{r!},\quad z,\lambda\in \mathbb {C}, \vert t \vert < 1. \end{aligned}$$

Proof

The proof runs parallel to that of Theorem 2.1 in [18]. We omit the details. □

Remark 4.1

By Remark 2.2, we can get all results of the sections (the weighted Caputo derivative and Generating functions), in the same way. Besides, these results become those in [12] when we consider the particular \(\omega(t,p,0)=e^{\frac{-p}{t(1-t)}}\) and \(\operatorname{Re}p>0\).

5 Further results and observations

Note that, if \(\omega^{(\alpha,\beta)}(t,p,s)=\frac{\Gamma (\beta)}{\Gamma(\alpha)\Gamma(\beta-\alpha)}\Gamma_{pt^{2}}(s)\), then \(B_{\omega}^{(\alpha,\beta)}\) becomes the following generalized gamma function (see [5], pp.32-33):

$$B^{(\alpha,\beta)}_{\omega}(\alpha-s,\beta-\alpha)=\frac{\Gamma(\beta )}{\Gamma(\alpha)\Gamma(\beta-\alpha)} \int_{0}^{1} t^{\alpha -s-1}(1-t)^{\beta-\alpha-1} \Gamma_{pt^{2}}(s)\,dt=\Gamma_{p}^{(\alpha,\beta)}(s), $$

where \(\min\{\operatorname{Re}y, \operatorname{Re}\alpha, \operatorname{Re}\beta ,\operatorname{Re}p, \operatorname{Re}s\}>0\), \(\Gamma_{p}(s)\) is Chaudhry’s gamma function [8] and \(\Gamma_{p}^{(\alpha,\beta)}(s)\) is defined in [5], p.33.

Now, we shall write some of the considered weighted functions in terms of the well-known Mittag-Leffler function [1]. We work with the generalization of the multivariable Mittag-Leffler function \(E_{(\rho_{j}),\lambda}^{(\gamma_{j}),(l_{j})}[z_{1},\ldots,z_{r}]\) introduced by Saxena et al. [20], p.547, Eq. (7.1) (see also [21], pp.2-3) and the generalized polynomials \(S_{n}^{m}[x]\) (see [2], p.1, Eq. (1)).

Corollary 5.1

If \(\omega(x_{1},z_{1},0)=S_{n_{1}}^{m_{1}}[x_{1}^{\lambda_{1}}(1-x_{1})^{\eta}]\exp\{ z_{1} (x_{1})^{\mu_{1}}(1-x_{1})^{\delta_{1}}\}\) and \(\eta,\lambda_{1},\delta_{1},\mu _{1}\in\mathbb{C}\); \(j\geq0\), \(\operatorname{Re}(a-m+n+k+\lambda_{1} j+\mu_{1} k_{1})>0\); \(\operatorname{Re}(d-a+m+nj+\delta_{1} k_{1})>0\), then

$$\begin{aligned} &F_{1}(a,b,c;d;x,y;\omega) \\ &\quad= \sum_{n,k=0}^{\infty}\sum _{j=0}^{[n_{1}/m_{1}]}\frac {(a)_{n+k}(b)_{n}(c)_{k}}{(a-m)_{n+k}}\frac{\Gamma(d)\Gamma(a-m+n+k+\lambda _{1} j)\Gamma(d-a+m+\lambda j)}{\Gamma(a-m)\Gamma(d-a+m)} \\ &\quad\quad{} \times\frac{(-n_{1})_{m_{1} j}}{j!} A_{n_{1},j}E_{(1,(\mu_{1}+\delta _{1})),1,d-a+m+\eta j+(a-m+n+k+\lambda_{1} j)}^{(1,a-m+n+k+\lambda_{1} j,d-a+m+\eta j),(1,\mu_{1},\delta_{1})}[z_{1}] \frac{x^{n}}{n!}\frac{y^{k}}{k!}, \end{aligned}$$

where \(\vert x \vert <1\), \(\vert y \vert <1\) and \(m<\operatorname{Re}a<\operatorname{Re}d\).

Proof

The desired equality holds by Corollary 3.13 in [21] (p.13) and Definition 2.2. □

Corollary 5.2

If \(\omega(x_{1},z_{1},0)=S_{n_{1}}^{m_{1}}[x_{1}^{\lambda_{1}}(1-x_{1})^{\eta }]E_{(\rho_{1}),\lambda}^{(\gamma_{1}),(l_{1})}[z_{1}(x_{1})^{\mu _{1}}(1-x_{1})^{\delta_{1}}]\) and \(\eta,\lambda,\lambda_{1},\gamma_{1},\rho _{1}, \delta_{1},\mu_{1}\in\mathbb{C}\); \(\operatorname{Re}\rho_{1}>0\); \(\operatorname{Re}\gamma_{1}>0\); \(j\geq0\), \(\operatorname{Re}(a-m+n+k+r+\lambda_{1} j+\mu_{1} k_{1})>0\); \(\operatorname{Re}(e-a+m+\eta j+\delta_{1} k_{1})>0\); \(l_{1}\in\mathbb {N}\); \(\lambda\notin\mathbb{Z}_{0}^{-}\), then

$$\begin{aligned} &F_{D,\omega}^{3}(a,b,c,d;e;x,y,z;\omega) \\ &\quad= \sum_{n,k,r=0}^{\infty}\sum _{j=0}^{[n_{1}/m_{1}]}\frac{(-n_{1})_{m_{1} j}}{j!}\frac{(a)_{n+k+r}(b)_{n}(c)_{k}(d)_{r}}{(a-m)_{n+k+r}} \frac{\Gamma (e)\Gamma(a-m+n+k+r+\lambda_{1} j)}{\Gamma(a-m)\Gamma(e-a+m)} \\ &\quad\quad{} \times\Gamma(e-a+m+\lambda j)A_{n_{1},j}E_{(\rho_{1},(\mu _{1}+\delta_{1})),\lambda,e-a+m+n_{1} j+(a-m+n+k+r+\lambda_{1} j)}^{(\gamma _{1},a-m+n+k+r+\lambda_{1} j,e-a+m+n_{1} j),(l_{1},\mu_{1},\delta_{1})}[z_{1}] \frac {x^{n}}{n!}\frac{y^{k}}{k!}\frac{z^{r}}{r!}, \end{aligned}$$

where \(\sqrt{ \vert x \vert }+\sqrt{ \vert y \vert }+\sqrt{ \vert z \vert }<1\) and \(m<\operatorname{Re}a<\operatorname{Re}e\).

Proof

The desired equality follows by a straightforward calculation using Corollary 3.10 in [21], p.12 and Definition 2.4. □

To present another application of the above results, we recall some well-known facts. We have Djrbashian’s Cauchy type kernel [22], p.76:

$$ C_{\omega}(z)=\sum_{k=0}^{+\infty} \frac{z^{k}}{\Delta_{k}},\quad\Delta _{0}=1,\Delta_{k}=k \int_{0}^{1} t^{k-1}\omega(t)\,dt, k \in \mathbb{N}. $$
(5.1)

One can see that \(C_{\omega}(z)\) is a holomorphic function in the unit disc (we denote this by \(\mathbb{D}=\{z\in\mathbb{C}: \vert z \vert <1\}\)) for any \(\omega(t)\in\Omega_{0}\) (where \(\Omega_{0}\) is the class defined in [22], p.76).

Remark 5.1

For the particular case of power functions \(\omega(x)=(1-x)^{\alpha}\), \(-1<\alpha<0\), \(C_{\omega}(z)\) is the \(1+\alpha\) order of the ordinary Cauchy kernel:

$$ C_{\omega}(z)=\frac{1}{(1-z)^{1+\alpha}}:=C_{\alpha}(z), \quad \text{and} \quad C_{1}(z)=\frac{1}{1-z},\quad z\in\mathbb{D}. $$
(5.2)

Definition 5.1

Let \(E\subset[0,2\pi]\) be a Borel measurable set (B-set) and \(\omega \in\Omega_{0}\). It is said that E is of positive ω-capacity \((C_{\omega}(E)>0)\) if there exists a nonnegative B-measure μ supported and finite on E and such that

$$S_{1}\equiv\lim_{r\to1-0} \max_{0\leq\varphi\leq2\pi} \int_{0}^{2\pi} \bigl\vert C_{\omega}\bigl(re^{i(\varphi-\theta)} \bigr) \bigr\vert \,d\mu(\theta)< +\infty. $$

If there is no such a measure, i.e. if \(S_{1}=\infty\) for any nonnegative B-measure, then E is said to be of zero ω-capacity \((C_{\omega}(E)=0)\).

Note that if we take \(\omega(x)=(1-x)^{\alpha}\) \((-1<\alpha <0)\), the last definition becomes that of the well-known Frostman α-capacity [22].

Corollary 5.3

If \(\sigma(\theta)\) is a function of bounded variation on \([0,2\pi]\), then the function

$$F_{\alpha}(z)=\frac{1}{2\pi} \int_{0}^{2\pi}{_{2}}F_{1} \bigl( \alpha+1,1;1;ze^{-i\theta };1 \bigr)\,d\sigma(\theta),\quad-1< \alpha< 0, \vert z \vert < 1, $$

has non-zero, finite nontangential boundary values \(F_{\alpha }(e^{i\varphi})\) at all points \(\varphi\in[0,2\pi]\), with a possible exception of a set \(S\subset[0,2\pi]\) of zero α-capacity.

Proof

First, one can see that

$${_{2}}F_{1} \bigl(\alpha+1,1;1;ze^{-i\theta};1 \bigr)= \frac{1}{(1-ze^{-i\theta})^{\alpha +1}}=\omega_{1} \bigl(ze^{-i\theta},\alpha \bigr), \quad \theta\in[0,2\pi]. $$

Besides, the function \(\omega(x)=(1-x)^{\alpha}\) \((-1<\alpha<0)\) belongs to the class \(\Omega_{0}\), and therefore \(C(ze^{-i\theta};\omega )\) becomes \(C(ze^{-i\theta};\alpha)=\omega_{1}(ze^{-i\theta},\alpha)\) \((\theta\in[0,2\pi])\) by Remark 5.1. Hence, by Theorem 2.5 in [22], p.112, we find that the function \(F_{\alpha}(z)\) has non-zero, finite nontangential boundary values at all points \(e^{i\varphi}\) \((\varphi\in[0,2\pi])\), except in a set \(S\subset[0,2\pi]\) of zero α-capacity. □

On the other hand, if f is a continuous function in \([0,b]\) and \(t\in[0,b]\), then \(D_{t}^{\mu,1}f(t)\) (the operator defined in equation (1.2)) becomes \({}_{0}^{RC} D_{t}^{\mu}f(t)\) (see [23], p.4, equation (10)), where \(m-1<\mu<m\), \(m\in\mathbb {N}\) (\(\Im \mu=0\)). Thus, if we consider the fractional problem about calculus of variations described in [23, 24] with our operator \(D_{z}^{\mu,\tau}f(z)\), under the conditions \(z\in\mathbb{R}\), \(m-1<\mu<m\) and \(m\in\mathbb{N}\). We find that the functional

$$I \bigl[p(\cdot) \bigr]= \int_{0}^{b} L \bigl(t,p(t),D_{t}^{\mu,1}p(t) \bigr)\,dt, $$

where \([0,b]\subset\mathbb{R}\), \(0< b\), \(0<\mu<1\), and the functions \(p(t)\) and the Lagrangian \(L:(t,p,vl)\to L(t,p,vl)\) are considered to be functions of class \(C^{2}\) (\(p(\cdot)\in C^{2}([0,b];\mathbb{R})\), \(L(\cdot,\cdot,\cdot)\in C^{2}([0,b]\times\mathbb{R}\times\mathbb {R};\mathbb{R})\)), satisfies the fractional Euler-Lagrange equation in the sense of Riesz-Caputo (see [23, 24]), i.e.

$$\partial_{2} L \bigl(t,p(t),D_{t}^{\mu,1}p(t) \bigr)- \frac{1}{\Gamma(1-\mu)}\frac {d}{dt} \int_{0}^{t} (t-\theta)^{-\mu} \bigl( \partial_{3} L \bigl(\theta,p(\theta ),D_{\theta}^{\mu,1}p( \theta) \bigr) \bigr)\,d\theta=0, $$

where \(\partial_{i} L\) is the partial derivative of L with respect to its ith argument (\(i=1,2,3\)) and \(p(\cdot)\) is an extremizer of the functional \(I [p(\cdot) ]\) for all \(t\in[0,b]\).

6 Conclusions

We hope to find some engineering applications related to our new results. Also, we analyze the possibilities to find solutions of partial differential equations or differential equations in terms of our results. Besides, we are trying to write the weighted hypergeomeric functions like Poisson integrals considering some special weights to find boundary values, factorizations of this functions and applications.