Here, we define a new extension of hypergeometric and confluent hypergeometric functions using the new extension of beta function.
Definition 5.1
The extension of hypergeometric function using the newly defined beta function involving Appell series \(F_{1}(\cdot)\) is
$$ F^{F_{1}}_{p,q}(\Psi _{1},\Psi _{2};\Psi _{3};w)=\sum_{n = 0}^{ \infty }( \Psi _{1})_{n} \frac{B_{p,q}^{F_{1}}(\Psi _{2}+n,\Psi _{3}-\Psi _{2})}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \frac{w^{n}}{n!}, $$
(44)
with \(\Re (\Psi _{3})>\Re (\Psi _{2})>0\) and \(\Re (p),\Re (q) \geq {0}\) and \(|w|<1\).
Definition 5.2
The extension of confluent hypergeometric function using the newly defined beta function involving Appell series \(F_{1}(\cdot)\) is
$$ \Phi ^{F_{1}}_{p,q}(\Psi _{2}; \Psi _{3};w)=\sum_{n = 0}^{ \infty } \frac{B_{p,q}^{F_{1}}(\Psi _{2}+n,\Psi _{3}-\Psi _{2})}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \frac{w^{n}}{n!}, $$
(45)
with \(\Re (\Psi _{3})>\Re (\Psi _{2})>0\) and \(\Re (p),\Re (q) \geq {0}\) and \(|w|<1\).
Similarly, we can define \(F^{F_{2}}_{p,q}(\Psi _{1},\Psi _{2};\Psi _{3};w)\), \(F^{F_{3}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)\), \(F^{F_{4}}_{p,q}(\Psi _{1},\Psi _{2}; \Psi _{3};w)\), \(\Phi ^{F_{2}}_{p,q}(\Psi _{2};\Psi _{3};w)\), \(\Phi ^{F_{3}}_{p,q}( \Psi _{2};\Psi _{3};w)\), and \(\Phi ^{F_{4}}_{p,q}(\Psi _{2};\Psi _{3};w)\).
Remark 5.1
When \(q=0\) and subsequently if \(p=1\), \(r=0\), \(a_{1},a_{2},a_{3}=1\), equations (44)–(45) reduce to the Gauss hypergeometric function (2) and confluent hypergeometric function (3), respectively.
5.1 Integral representation
Theorem 5.1
The extended hypergeometric function has the following integral representation:
$$ \begin{aligned}[b] F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}& \frac{1}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{1}t^{\Psi _{2}-1}(1-t)^{ \Psi _{3}-\Psi _{2}-1}(1-tw)^{-\Psi _{1}} \\ &{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p}{t^{r}}, \frac{q}{(1-t)^{r}}} \biggr)}\,dt, \end{aligned} $$
(46)
where \(\Re (a_{1}),\Re (a_{2}),\Re (a_{2}^{\prime }),\Re (a_{3})>0\), \(\Re (p),\Re (q) \geq {0}\), \(r \geq {0}\), \(\Re (\Psi _{3})>\Re (\Psi _{2})>0\), \(|w|<1\), and \(|\arg {(1-t)}|<\pi \).
Proof
By definition (44),
$$ F^{F_{1}}_{p,q}(\Psi _{1},\Psi _{2};\Psi _{3};w)=\sum_{n = 0}^{ \infty }( \Psi _{1})_{n} \frac{B_{p,q}^{F_{1}}(\Psi _{2}+n,\Psi _{3}-\Psi _{2})}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \frac{w^{n}}{n!}. $$
By definition (20) of the extended beta function
$$\begin{aligned}& \begin{aligned} F^{F_{1}}_{p,q}(\Psi _{1},\Psi _{2};\Psi _{3};w)={}& \frac{1}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{1}t^{\Psi _{2}+n-1}(1-t)^{ \Psi _{3}-\Psi _{2}-1} \sum_{n=0}^{\infty }(\Psi _{1})_{n} \\ &{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p}{t^{r}}, \frac{q}{(1-t)^{r}}} \biggr)}\,dt\frac{w^{n}}{n!}. \end{aligned} \\& \text{Using } \sum_{n=0}^{\infty } \frac{(\Psi _{1})_{n}(tw)^{n}}{n!}=(1-t)^{-\Psi _{1}}, \text{we get the desired result (46)}. \end{aligned}$$
□
Theorem 5.2
The extended confluent hypergeometric function has the following integral representation:
$$\begin{aligned}& \begin{aligned}[b] \Phi ^{F_{1}}_{p,q}( \Psi _{2};\Psi _{3};w)={}& \frac{1}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{1}t^{\Psi _{2}-1}(1-t)^{ \Psi _{3}-\Psi _{2}-1} \exp {(wt)} \\ &{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p}{t^{r}}, \frac{q}{(1-t)^{r}}} \biggr)}\,dt, \end{aligned} \end{aligned}$$
(47)
$$\begin{aligned}& \begin{aligned}[b] \Phi ^{F_{1}}_{p,q}( \Psi _{2};\Psi _{3};w)={}& \frac{\exp {(w)}}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{1}t^{ \Psi _{2}-1}(1-t)^{\Psi _{3}-\Psi _{2}-1} \exp {(-wt)} \\ &{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p}{t^{r}}, \frac{q}{(1-t)^{r}}} \biggr)}\,dt, \end{aligned} \end{aligned}$$
(48)
where \(\Re (a_{1}),\Re (a_{2}),\Re (a_{2}^{\prime }),\Re (a_{3})>0\), \(\Re (p),\Re (q) \geq {0}\), \(r \geq {0}\), \(\Re (\Psi _{3})>\Re (\Psi _{2})>0\), and \(|w|<1\).
Proof
By definition (45)
$$ \Phi ^{F_{1}}_{p,q}(\Psi _{2};\Psi _{3};w)=\sum_{n = 0}^{ \infty } \frac{B_{p,q}^{F_{1}}(\Psi _{2}+n,\Psi _{3}-\Psi _{2})}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \frac{w^{n}}{n!}. $$
By definition (20) of the extended beta function
$$\begin{aligned}& \begin{aligned} \Phi ^{F_{1}}_{p,q}(\Psi _{2};\Psi _{3};w)={}& \frac{1}{B(\Psi _{2},\Psi _{3}-\Psi _{2})}\sum _{n=0}^{\infty } \int _{0}^{1}t^{\Psi _{2}+n-1}(1-t)^{\Psi _{3}-\Psi _{2}-1} \\ &{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p}{t^{r}}, \frac{q}{(1-t)^{r}}} \biggr)}\,dt\frac{w^{n}}{n!}. \end{aligned} \\& \text{Using } \sum_{n=0}^{\infty } \frac{(tw)^{n}}{n!}= \exp {(wt)}, \text{we get the desired result (47)}. \end{aligned}$$
Replacing t by \((1-t)\) in equation (47), we get result (48). □
Theorem 5.3
The following integral representations for the extended hypergeometric function hold true:
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{i})&\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)= \frac{1}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{\infty }a^{\Psi _{2}-1}(1+a)^{\Psi _{1}-\Psi _{3}} \bigl(1+a(1-w)\bigr)^{- \Psi _{1}} \\ &\hphantom{\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}}{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p(1+a)^{r}}{a^{r}},q(1+a)^{r}} \biggr)}\,da, \end{aligned} \end{aligned}$$
(49)
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{ii}) &\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)= \frac{2}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{\frac{\pi }{2}}\cos ^{2\Psi _{2}-1}\theta \sin ^{2\Psi _{3}-2 \Psi _{2}-1}\theta \\ &\hphantom{\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}}{}\times\bigl(1-\cos ^{2}\theta {w}\bigr)^{-\Psi _{1}} \\ &\hphantom{\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}}{}\times{{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p}{\cos ^{2r}\theta },\frac{q}{\sin ^{2r}\theta }} \biggr)}\,d\theta , \end{aligned} \end{aligned}$$
(50)
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{iii})&\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)= \frac{2^{1+\Psi _{1}-\Psi _{3}}}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{-1}^{1}(1-a)^{\Psi _{3}-\Psi _{2}-1} \bigl(2-w(1+a)\bigr)^{-\Psi _{1}} \\ &\hphantom{\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}}{}\times (1+a)^{\Psi _{2}-1} \\ &\hphantom{\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}}{}\times{{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{2^{r}p}{(1+a)^{r}},\frac{2^{r}q}{(1-a)^{r}}} \biggr)}\,da , \end{aligned} \end{aligned}$$
(51)
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{iv})&\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)= \frac{(c-u)^{1+\Psi _{1}-\Psi _{3}}}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{u}^{c}(a-u)^{\Psi _{2}-1} \bigl((c-u)-w(a-u) \bigr)^{-\Psi _{1}} \\ &\hphantom{\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}}{}\times (c-a)^{\Psi _{3}-\Psi _{2}-1} \\ &\hphantom{\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}}{}\times{{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p(c-u)^{r}}{(a-u)^{r}},\frac{q(c-u)^{r}}{(c-a)^{r}}} \biggr)}\,da , \end{aligned} \end{aligned}$$
(52)
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{v})&\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)= \frac{1}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{\frac{\pi }{4}}\tanh ^{2\Psi _{2}-2}\theta \operatorname{sech}^{2\Psi _{3}-2 \Psi _{2}}\theta \\ &\hphantom{\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}}{}\times \bigl(1-\tanh ^{2}\theta {w}\bigr)^{-\Psi _{1}} \\ &\hphantom{\quad F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)={}}{}\times{{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p}{\tanh ^{2r}\theta },\frac{q}{\operatorname{sech}^{2r}\theta }} \biggr)}\,d\theta . \end{aligned} \end{aligned}$$
(53)
Proof
Substitute \(t=\frac{a}{1+a}\), \(t=\cos ^{2}\theta \), \(t=\frac{1+a}{2}\), \(t= \frac{a-u}{c-u}\), and \(t=\tanh ^{2}\theta \) in equation (46) to get equations (49)–(53). □
Theorem 5.4
The following integral representations for the extended confluent hypergeometric function hold true:
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{i})&\quad \Phi ^{F_{1}}_{p,q}( \Psi _{2};\Psi _{3};w)= \frac{1}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{\infty }a^{\Psi _{2}-1}(1+a)^{- \Psi _{3}} \exp { \biggl(\frac{wa}{1+a} \biggr)} \\ &\hphantom{\quad \Phi ^{F_{1}}_{p,q}( \Psi _{2};\Psi _{3};w)={}}{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p(1+a)^{r}}{a^{r}},q(1+a)^{r}} \biggr)}\,da, \end{aligned} \end{aligned}$$
(54)
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{ii})&\quad \Phi ^{F_{1}}_{p,q}(\Psi _{2};\Psi _{3};w)= \frac{2}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{\frac{\pi }{2}} \cos ^{2\Psi _{2}-1}\theta \sin ^{2\Psi _{3}-2\Psi _{2}-1}\theta \exp {\bigl(w \cos ^{2}\theta \bigr)} \\ &\hphantom{\quad \Phi ^{F_{1}}_{p,q}(\Psi _{2};\Psi _{3};w)={}}{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p}{\cos ^{2r}\theta },\frac{q}{\sin ^{2r}\theta }} \biggr)}\,d\theta , \end{aligned} \end{aligned}$$
(55)
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{iii})&\quad \Phi ^{F_{1}}_{p,q}(\Psi _{2};\Psi _{3};w)= \frac{2^{1-\Psi _{3}}}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{-1}^{1}(1+a)^{ \Psi _{2}-1}(1-a)^{\Psi _{3}-\Psi _{2}-1} \exp \biggl({ \frac{w(1+a)}{2}} \biggr) \\ &\hphantom{\quad \Phi ^{F_{1}}_{p,q}(\Psi _{2};\Psi _{3};w)={}}{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{2^{r}p}{(1+a)^{r}},\frac{2^{r}q}{(1-a)^{r}}} \biggr)}\,da, \end{aligned} \end{aligned}$$
(56)
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{iv})&\quad \Phi ^{F_{1}}_{p,q}(\Psi _{2};\Psi _{3};w)= \frac{(c-u)^{1-\Psi _{3}}}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{u}^{c}(a-u)^{ \Psi _{2}-1}(c-a)^{\Psi _{3}-\Psi _{2}-1} \exp { \biggl( \frac{w(a-u)}{(c-u)} \biggr)} \\ &\hphantom{\quad \Phi ^{F_{1}}_{p,q}(\Psi _{2};\Psi _{3};w)={}}{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p(c-u)^{r}}{(a-u)^{r}},\frac{q(c-u)^{r}}{(c-a)^{r}}} \biggr)}\,da, \end{aligned} \end{aligned}$$
(57)
$$\begin{aligned}& \begin{aligned}[b] (\mathrm{v})&\quad \Phi ^{F_{1}}_{p,q}( \Psi _{2};\Psi _{3};w)= \frac{1}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \int _{0}^{\frac{\pi }{4}} \tanh ^{2\Psi _{2}-2}\theta \operatorname{sech}^{2\Psi _{3}-2\Psi _{2}}\theta \exp {\bigl(w \tanh ^{2}\theta \bigr)} \\ &\hphantom{\quad \Phi ^{F_{1}}_{p,q}( \Psi _{2};\Psi _{3};w)={}}{}\times {{F_{1}} \biggl({a_{1},a_{2},a_{2}^{\prime };a_{3}; \frac{p}{\tanh ^{2r}\theta },\frac{q}{\operatorname{sech}^{2r}\theta }} \biggr)}\,d\theta . \end{aligned} \end{aligned}$$
(58)
Proof
Substitute \(t=\frac{a}{1+a}\), \(t=\cos ^{2}\theta \), \(t=\frac{1+a}{2}\), \(t= \frac{a-u}{c-u}\), and \(t=\tanh ^{2}\theta \) in equation (47) to get equations (54)–(58). □
5.2 Differentiation formula
Theorem 5.5
The following differentiation formulas for the extended hypergeometric and confluent hypergeometric function hold true:
$$ \frac{d^{n}}{dw^{n}} \bigl\{ F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w) \bigr\} = \frac{(\Psi _{1})_{n}(\Psi _{2})_{n}}{(\Psi _{3})_{n}}F^{F_{1}}_{p,q}( \Psi _{1}+n, \Psi _{2}+n;\Psi _{3}+n;w) $$
(59)
and
$$ \frac{d^{n}}{dw^{n}} \bigl\{ \Phi ^{F_{1}}_{p,q}( \Psi _{2};\Psi _{3};w) \bigr\} =\frac{(\Psi _{2})_{n}}{(\Psi _{3})_{n}} \Phi ^{F_{1}}_{p,q}( \Psi _{2}+n;\Psi _{3}+n;w). $$
(60)
Proof
Differentiating equation (44) with respect to w, we get
$$\begin{aligned} \frac{d}{dw} \bigl\{ F^{F_{1}}_{p,q}(\Psi _{1},\Psi _{2};\Psi _{3};w) \bigr\} =\sum _{n = 1}^{\infty }(\Psi _{1})_{n} \frac{B_{p,q}^{F_{1}}(\Psi _{2}+n,\Psi _{3}-\Psi _{2})}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \frac{w^{n-1}}{(n-1)!}. \end{aligned}$$
Replacing n by \(n+1\)
$$ \frac{d}{dw} \bigl\{ F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w) \bigr\} = \sum_{n = 0}^{\infty }(\Psi _{1})_{n+1} \frac{B_{p,q}^{F_{1}}(\Psi _{2}+n+1,\Psi _{3}-\Psi _{2})}{B(\Psi _{2},\Psi _{3}-\Psi _{2})} \frac{w^{n}}{n!}. $$
(61)
Using \(B(h,k-h)=\frac{k}{h}B(h+1,k-h)\) in equation (61), we get
$$\begin{aligned}& \frac{d}{dw} \bigl\{ F^{F_{1}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w) \bigr\} = \sum_{n = 0}^{\infty }(\Psi _{1})_{n+1} \biggl( \frac{\Psi _{2}}{\Psi _{3}} \biggr) \frac{B_{p,q}^{F_{1}}(\Psi _{2}+n+1,\Psi _{3} -\Psi _{2})}{B(\Psi _{2}+1,\Psi _{3}-\Psi _{2})} \frac{w^{n}}{n!} \\& \begin{aligned} \quad \implies\quad \frac{d}{dw} \bigl\{ F^{F_{1}}_{p,q}(\Psi _{1},\Psi _{2}; \Psi _{3};w) \bigr\} &=\sum _{n = 0}^{\infty }(\Psi _{1})_{n} \biggl(\frac{\Psi _{1}\Psi _{2}}{\Psi _{3}} \biggr) \frac{B_{p,q}^{F_{1}}(\Psi _{2}+n+1,\Psi _{3}-\Psi _{2})}{B(\Psi _{2}+1,\Psi _{3}-\Psi _{2})} \frac{w^{n}}{n!} \\ &=\frac{\Psi _{1}\Psi _{2}}{\Psi _{3}}F^{F_{1}}_{p,q}(\Psi _{1}+1, \Psi _{2}+1;\Psi _{3}+1;w). \end{aligned} \end{aligned}$$
(62)
Again differentiating equation (62) with respect to w, we get
$$ \frac{d^{2}}{dw^{2}} \bigl\{ F^{F_{1}}_{p,q}(\Psi _{1},\Psi _{2};\Psi _{3};w) \bigr\} = \frac{\Psi _{1}(\Psi _{1}+1)\Psi _{2}(\Psi _{2}+1)}{\Psi _{3}(\Psi _{3}+1)}F^{F_{1}}_{p,q}( \Psi _{1}+2, \Psi _{2}+2;\Psi _{3}+2;w). $$
Continuing like this, n times, we get the desired result (59).
We can prove result (60) in a similar way. □
Similarly, we can prove the above results for \(F^{F_{2}}_{p,q}(\Psi _{1},\Psi _{2};\Psi _{3};w)\), \(F^{F_{3}}_{p,q}( \Psi _{1},\Psi _{2};\Psi _{3};w)\), \(F^{F_{4}}_{p,q}(\Psi _{1},\Psi _{2};\Psi _{3};w)\), \(\Phi ^{F_{2}}_{p,q}( \Psi _{2};\Psi _{3};w)\), \(\Phi ^{F_{3}}_{p,q}(\Psi _{2};\Psi _{3};w)\), and \(\Phi ^{F_{4}}_{p,q}(\Psi _{2};\Psi _{3};w)\).