2 A New Generalization of the Pochhammer Symbol
In this section, we denote a new generalization of Pochhammer symbol (5) . Also, we give some useful properties.
Definition 1
Let λ , μ ∈ ℂ and ℜ(p ) > 0, ℜ(q ) > 0, ℜ(κ ) > 0, ℜ(μ ) > 0, the generalization of the extended Pochhammer symbol (λ ; p ,q ;κ , μ )ν is given by
(5) ![]()
( λ ; p , q ; κ , μ ) ν : = { Γ p , q ( κ , μ ) ( λ + ν ) Γ ( λ ) , ℜ ( p ) > 0 , ℜ ( q ) > 0 , ℜ ( κ ) > 0 , ℜ ( μ ) > 0 , ( λ ) ν , p = 1 , q = 0 , κ = 1 , μ = 0 , (\lambda ;p,q;\kappa, \mu )_\nu : = \left\{ {\matrix{ {{{\Gamma _{p,q}^{(\kappa, \mu )} (\lambda + \nu )} \over {\Gamma (\lambda )}}} \hfill & {, \, \Re (p) > 0, \, \Re (q) > 0, \, \Re (\kappa ) > 0, \, \Re (\mu ) > 0,} \hfill \cr \, \, \, \, \, \, \, {(\lambda )_\nu } \hfill & {, \, p = 1, \, q = 0, \, \kappa = 1, \, \mu = 0,} \hfill \cr } } \right.
where
Γ p , q ( κ , μ ) \Gamma _{p,q}^{(\kappa, \mu )}
is the generalization of the extended gamma function (4) [31
Theorem 1
For the generalization of the Pochhammer symbol (5) following integral representation holds true: (6) ![]()
( λ ; p , q ; κ , μ ) ν : = 1 Γ ( λ ) ∫ 0 ∞ t λ + ν − 1 exp ( − t κ p − q t μ ) dt ( ℜ ( p ) > 0 , ℜ ( q ) > 0 , ℜ ( κ ) > 0 , ℜ ( μ ) > 0 ) . \matrix{ {(\lambda ;p,q;\kappa, \mu )_\nu : = {1 \over {\Gamma (\lambda )}} \, \int_0^\infty \, t^{\lambda + \nu - 1} \, \exp \left( { - {{t^\kappa } \over p} - {q \over {t^\mu }}} \right)dt} \cr {(\Re (p) > 0, \, \Re (q) > 0, \, \Re (\kappa ) > 0, \, \Re (\mu ) > 0).}}
Proof
Using the equality (4) in the definition of the (5) , we get the desired result (6) .
Theorem 2
Let λ, m, n ∈ ℂ. Then ,
(7) ![]()
( λ ; p , q ; κ , μ ) n + m : = ( λ ) n ( λ + n ; p , q ; κ , μ ) m (\lambda ;p,q;\kappa, \mu )_{n + m} : = (\lambda )_n (\lambda + n;p,q;\kappa, \mu )_m
Proof
From the equations (1) and (5) , we obtain that
(8) ![]()
( λ ; p , q ; κ , μ ) m + n : = Γ p , q ( κ , μ ) ( λ + m + n ) Γ ( λ ) = Γ ( λ + n ) Γ ( λ + n ) Γ p , q ( κ , μ ) ( λ + m + n ) Γ ( λ ) = ( λ ) n ( λ + n ; p , q ; κ , μ ) m \matrix{ {(\lambda ;p,q;\kappa, \mu )_{m + n} } \hfill & {: = {{\Gamma _{p,q}^{(\kappa, \mu )} (\lambda + m + n)} \over {\Gamma (\lambda )}} = {{\Gamma (\lambda + n)} \over {\Gamma (\lambda + n)}}{{\Gamma _{p,q}^{(\kappa, \mu )} (\lambda + m + n)} \over {\Gamma (\lambda )}}} \hfill \cr {} \hfill & { = (\lambda )_n (\lambda + n;p,q;\kappa, \mu )_m }}
By appealing the well-known properties of the classical Pochhammer symbol in (7) following features of generalization of the Pochhammer symbol can be easily obtained.
Corollary 3
Let k,l,m,n ∈ ℕ0 and N ∈ ℕ. Then ,
![]()
( λ ; p , q ; κ , μ ) m + n + l : = ( λ ) m ( λ + m ) n ( λ + m + n ; p , q ; κ , μ ) l ( λ ; p , q ; κ , μ ) m − n + l : = ( − 1 ) n ( λ ) m ( 1 − λ − m ) n ( λ + m − n ; p , q ; κ , μ ) l ( λ ; p , q ; κ , μ ) 2 m + l : = 2 2 m ( λ 2 ) m ( λ + 1 2 ) m ( λ + 2 m ; p , q ; κ , μ ) l ( λ ; p , q ; κ , μ ) N m + l : = N N m ( λ N ) m ( λ + 1 N ) m … ( λ + N − 1 N ) m ( λ + N m ; p , q ; κ , μ ) l ( λ + n ; p , q ; μ ) n + 1 : = ( λ + n ) n ( λ + 2 n ; p , q ; κ , μ ) l = ( λ ) 2 n ( λ ) n ( λ + 2 n ; p , q , κ , μ ) l ( λ + m ; p , q ; κ , μ ) n + l : = ( λ ) n ( λ + n ) m ( λ ) m ( λ + m + n ; p , q ; κ , μ ) l ( λ + km ; p , q ; κ , μ ) kn + l : = ( λ ) km + kn ( λ ) km ( λ + km + kn ; p , q ; κ , μ ) l ( λ − n ; p , q ; κ , μ ) n + 1 : = ( − 1 ) n ( 1 − λ ) n ( λ ; p , q ; κ , μ ) l ( λ − m ; p , q ; κ , μ ) n + 1 : = ( 1 − λ ) m ( λ ) n ( 1 − λ − n ) m ( λ + n − ; p , q , κ , μ ) l ( λ − km ; p , q ; κ , μ ) kn + 1 : = ( − 1 ) km ( λ ) kn − km ( 1 − λ ) km ( λ + kn − km ; p , q , κ , μ ) l ( λ + m ; p , q ; κ , μ ) n − m + l : = ( λ ) n ( λ ) m ( λ + n + ; p , q ; κ , μ ) l ( λ − m ; p , q ; κ , μ ) n − m + 1 : = ( − 1 ) m ( λ ) n ( 1 − λ ) m ( 1 − λ − n ) 2 m ( λ + n − 2 m ; p , q , κ , μ ) l ( − λ ; p , q ; κ , μ ) n + 1 : = ( − 1 ) n ( λ − n + 1 ) ( − λ + n ; p , q , κ , μ ) l \eqalign{ & \left( {\lambda ;p,q;\kappa ,\mu } \right)_{m + n + l} : = \left( \lambda \right)_m \left( {\lambda + m} \right)_n \left( {\lambda + m + n;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda ;p,q;\kappa ,\mu } \right)_{m - n + l} : = {{\left( { - 1} \right)^n \left( \lambda \right)_m } \over {\left( {1 - \lambda - m} \right)_n }}\left( {\lambda + m - n;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda ;p,q;\kappa ,\mu } \right)_{2m + l} : = 2^{2m} \left( {{\lambda \over 2}} \right)_m \left( {{{\lambda + 1} \over 2}} \right)_m \left( {\lambda + 2m;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda ;p,q;\kappa ,\mu } \right)_{{\rm {N}}m + l} : = {\rm {N}}^{{\rm {N}}m} \left( {{\lambda \over {\rm {N}}}} \right)_m \left( {{{\lambda + 1} \over {\rm {N}}}} \right)_m \ldots \left( {{{\lambda + {\rm {N}} - 1} \over {\rm {N}}}} \right)_m \left( {\lambda + {\rm {N}}m;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda + n;p,q;\mu } \right)_{n + 1} : = \left( {\lambda + n} \right)_n \left( {\lambda + 2n;p,q;\kappa ,\mu } \right)_l = {{\left( \lambda \right)_{2n} } \over {\left( \lambda \right)_n }}\left( {\lambda + 2n;p,q,\kappa ,\mu } \right)_l \cr & \left( {\lambda + m;p,q;\kappa ,\mu } \right)_{n + l} : = {{\left( \lambda \right)_n \left( {\lambda + n} \right)_m } \over {\left( \lambda \right)_m }}\left( {\lambda + m + n;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda + km;p,q;\kappa ,\mu } \right)_{kn + l} : = {{\left( \lambda \right)_{km + kn} } \over {\left( \lambda \right)_{km} }}\left( {\lambda + km + kn;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda - n;p,q;\kappa ,\mu } \right)_{n + 1} : = \left( { - 1} \right)^n \left( {1 - \lambda } \right)_n \left( {\lambda ;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda - m;p,q;\kappa ,\mu } \right)_{n + 1} : = {{\left( {1 - \lambda } \right)_m \left( \lambda \right)_n } \over {\left( {1 - \lambda - n} \right)_m }}\left( {\lambda + n - ;p,q,\kappa ,\mu } \right)_l \cr & \left( {\lambda - km;p,q;\kappa ,\mu } \right)_{kn + 1} : = \left( { - 1} \right)^{km} \left( \lambda \right)_{kn - km} \left( {1 - \lambda } \right)_{km} \left( {\lambda + kn - km;p,q,\kappa ,\mu } \right)_l \cr & \left( {\lambda + m;p,q;\kappa ,\mu } \right)_{n - m + l} : = {{\left( \lambda \right)_n } \over {\left( \lambda \right)_m }}\left( {\lambda + n + ;p,q;\kappa ,\mu } \right)_l \cr & \left( {\lambda - m;p,q;\kappa ,\mu } \right)_{n - m + 1} : = \left( { - 1} \right)^m {{\left( \lambda \right)_n \left( {1 - \lambda } \right)_m } \over {\left( {1 - \lambda - n} \right)_{2m} }}\left( {\lambda + n - 2m;p,q,\kappa ,\mu } \right)_l \cr & \left( { - \lambda ;p,q;\kappa ,\mu } \right)_{n + 1} : = \left( { - 1} \right)^n \left( {\lambda - n + 1} \right)\left( { - \lambda + n;p,q,\kappa ,\mu } \right)_l .}
Remark 1
Taking p = κ = μ = 1 in the Corollary 3 , it is easily seen that the special case of extended Pochhammer symbol [29 ].
3 A New Generalization of the extended hypergeometric function
According to the generalization of the extended Pochhammer symbol (λ ; p,q ; κ, μ )n (n ∈ ℕ0 ), a generalization of the extended hypergeometric function r F s of r numerator parameters a 1 ,⋯,a r and s denominator parameters b 1 ,⋯,b s can be given as follows:
(9) ![]()
r F s [ ( a 1 ; p , q ; κ , μ ) , a 2 , ⋯ , a r b 1 , b 2 , ⋯ , b s ; z ] : = ∑ n = 0 ∞ ( a 1 ; p , q ; κ , μ ) n ( a 2 ) n ⋯ ( a r ) n ( b 1 ) n ⋯ ( b s ) n ⋅ z n n ! , _r F_s \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right]: = \sum\limits_{n = 0}^\infty {{(a_1 ;p,q;\kappa, \mu )_n (a_2 )_n \cdots (a_r )_n } \over {(b_1 )_n \cdots (b_s )_n }} \cdot {{z^n } \over {n!}},
on condition that the series on the right-hand side converges, it making sense that aj ∈ ℂ ( j = 1,..., r ) and
b j ∈ ℂ \ ℤ 0 − ( j = 1 , ... , s ; ℤ 0 − = { 0 , − 1 , − 2 , ... } ) b_j \in {\mathbb{C}} \backslash {\mathbb{Z}}_0^ - (j = 1,...,s;\, \, {\mathbb{Z}}_0^ - = \{ 0, - 1, - 2,...\} )
.
Particularly, the corresponding generalization of the extended confluent hypergeometric function
Φ p , q κ , μ \Phi _{p,q}^{\kappa, \mu }
and the Gauss hypergeometric function
F p , q κ , μ F_{p,q}^{\kappa, \mu }
are given by
(10) ![]()
Φ p , q κ , μ ( a ; b ; z ) : = ∑ n = 0 ∞ ( a ; p , q ; κ , μ ) n ( b ) n ⋅ z n n ! \Phi _{p,q}^{\kappa, \mu } (a;b;z): = \, \sum\limits_{n = 0}^\infty \, {{(a;p,q;\kappa, \mu )_n } \over {(b)_n }} \cdot {{z^n } \over {n!}}
and
(11) ![]()
F p , q κ , μ ( a , b , c ; z ) : = ∑ n = 0 ∞ ( a ; p , q ; κ , μ ) n ( b ) n ( c ) n ⋅ z n n ! , F_{p,q}^{\kappa, \mu } (a,b,c;z): = \sum\limits_{n = 0}^\infty \, {{(a;p,q;\kappa, \mu )_n (b)_n } \over {(c)_n }} \cdot {{z^n } \over {n!}},
respectively.
Theorem 4
The following integral representation holds true: (12) ![]()
r F s [ ( a 1 ; p , q ; κ , μ ) , a 2 , ⋯ , a r b 1 , b 2 , ⋯ , b s ; z ] : = 1 Γ ( a 1 ) ∫ 0 ∞ t a 1 − 1 exp ( − t κ p − q t μ ) × r − 1 F s [ a 2 , ⋯ , a r b 1 , b 2 , ⋯ , b s ; zt ] dt , ( ℜ ( p ) > 0 , ℜ ( q ) > 0 , ℜ ( κ ) > 0 , ℜ ( μ ) > 0 ; ℜ ( b s ) > ℜ ( a r ) > 0 , ) . \eqalign{ & _r F_s \left[ {(a_1 ;\,p,q;\,\kappa, \,\mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right] \cr & : = {1 \over {\Gamma (a_1 )}}\int\limits_0^\infty t^{a_1 - 1} \exp \left( { - {{t^\kappa } \over p} - {q \over {t^\mu }}} \right) \times _{r - 1} F_s \left[ {a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;zt} \right]dt, \cr & \, \, \, \, \, \, \, \, (\Re (p) > 0, \, \, \,\Re (q) > 0, \, \,\Re (\kappa ) > 0, \, \,\Re (\mu ) > 0; \, \,\Re (b_s ) > \Re (a_r ) > 0,).}
Proof
Using the integral representation given by (6) in the definition (9) , we led to desired result (12) .
Theorem 5
The following integral representation holds true: (13) ![]()
r F s [ ( a 1 ; p , q ; κ , μ ) , a 2 , ⋯ , a r b 1 , b 2 , ⋯ , b s ; z ] : = 1 B ( a r , b s − a r ) ∫ 0 1 t a r − 1 ( 1 − t ) b s − a r − 1 × r − 1 F s − 1 [ ( a 1 ; p , q ; κ , μ ) , a 2 , ⋯ , a r − 1 b 1 , b 2 , ⋯ , b s − 1 ; zt ] dt , ( ℜ ( p ) > 0 , ℜ ( q ) > 0 , ℜ ( κ ) > 0 , ℜ ( μ ) > 0 ; ℜ ( b s ) > ℜ ( a r ) > 0 ) . \eqalign{ & \matrix{ {_r F_s \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right]} \hfill & {: = {1 \over {B(a_r, b_s - a_r )}}\int\limits_0^1 \, t^{a_r - 1} (1 - t)^{b_s - a_r - 1} } \hfill \cr {} \hfill & { \times \, _{r - 1} F_{s - 1} \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_{r - 1} \, b_1, b_2, \cdots, b_{s - 1} \, ;zt} \right]dt,} \hfill \cr } \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\Re (p) > 0, \, \,\Re (q) > 0, \, \,\Re (\kappa ) > 0,\, \, \Re (\mu ) > 0; \, \,\Re (b_s ) > \Re (a_r ) > 0).}
Proof
The classical Beta function B (α,β ) defined by [1 , 7 , 9 , 10 , 18 , 22 , 23 , 26 , 29 , 30 , 33 ],
(14) ![]()
B ( α , β ) = { ∫ 0 1 t α − 1 ( 1 − t ) β − 1 dt ( min { ℝ ( α ) , ℝ ( β ) } > 0 ) Γ ( α ) Γ ( β ) Γ ( α + β ) ( α , β ∈ ℂ \ ℤ 0 − ) . B(\alpha, \beta ) = \left\{ {\matrix{ {\int\limits_0^1 t^{\alpha - 1} (1 - t)^{\beta - 1} dt} \hfill & {(\min \{ {\mathbb{R}}(\alpha ),{\mathbb{R}}(\beta )\} > 0)} \hfill \cr {{{\Gamma (\alpha )\Gamma (\beta )} \over {\Gamma (\alpha + \beta )}}} \hfill & {(\alpha, \beta \in {\mathbb{C}} \backslash {\mathbb{Z}}_0^ - ).}} } \right.
Also, we have the following equation
(15) ![]()
( a r ) n ( b s ) n = 1 B ( a r , b s − a r ) ∫ 0 1 t a r + n − 1 ( 1 − t ) b s − a r − 1 dt , ( ℝ ( b s ) > ℝ ( a r ) > 0 ; n ∈ ℕ 0 ) \matrix{ {{{(a_r )_n } \over {(b_s )_n }} = {1 \over {B(a_r,\ b_s - a_r )}}\int\limits_0^1 t^{a_r + n - 1} (1 - t)^{b_s - a_r - 1} dt,} \cr {({\mathbb{R}}(b_s ) > {\mathbb{R}}(a_r ) > 0;n \in {\mathbb{N}}_0 )}}
Using the equalities (14) , (15) in the generization of the extended hypergeometric function (9) , we get the desired result (13) .
Corollary 6
Each of the integral representations hold true: (16) ![]()
Φ p , q κ , μ ( a ; b ; z ) = 1 Γ ( a ) ∫ 0 ∞ t a − 1 exp ( − t κ p − q t μ ) 0 F 1 ( − ; b ; zt ) dt , \Phi _{p,q}^{\kappa, \mu } (a;b;z) = {1 \over {\Gamma (a)}}\int\limits_0^\infty t^{a - 1} \exp \left( { - {{t^\kappa } \over p} - {q \over {t^\mu }}} \right) \, _0 F_1 ( - ;b;zt)dt, (17) ![]()
F p , q κ , μ ( a , b , c ; z ) = 1 Γ ( a ) ∫ 0 ∞ t a − 1 exp ( − t κ p − q t μ ) 1 F 1 ( b ; c ; zt ) dt , F_{p,q}^{\kappa, \mu } (a,b,c;z) = {1 \over {\Gamma (a)}}\int\limits_0^\infty t^{a - 1} \exp \left( { - {{t^\kappa } \over p} - {q \over {t^\mu }}} \right) \, _1 F_1 (b;c;zt)dt, and (18) ![]()
F p , q κ , μ ( a , b , c ; z ) = 1 B ( b , c − b ) ∫ 0 1 t b − 1 ( 1 − t ) c − b − 1 1 F 0 ( ( a ; p , q ; κ , μ ) ; − ; zt ) dt , F_{p,q}^{\kappa, \mu } (a,b,c;z) = {1 \over {B(b,c - b)}} \, \int\limits_0^1 t^{b - 1} (1 - t)^{c - b - 1} \, _1 F_0 ((a;p,q;\kappa, \mu ); - ;zt)dt, on condition that the integrals involved are convergent.
Theorem 7
The following derivative formula holds true: (19) ![]()
d n dz n { r F s [ ( a 1 ; p , q ; κ , μ ) , a 2 , ⋯ , a r b 1 , b 2 , ⋯ , b s ; z ] } : = ( a 1 ) n ( a 2 ) n ⋯ ( a r ) n ( b 1 ) n ⋯ ( b s ) n × r F s [ ( a 1 + n ; p , q ; κ , μ ) , a 2 + n , ⋯ , a r + n b 1 + n , b 2 + n , ⋯ , b s + n ; z ] ( ℜ ( p ) > 0 , ℜ ( q ) > 0 , ℜ ( κ ) > 0 , ℜ ( μ ) > 0 ; n ∈ ℕ 0 ) . \eqalign{ & {{{d^n } \over {dz^n }}\left\{ {_r F_s \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right]} \right\}\,:\, = \,{{(a_1 )_n (a_2 )_n \cdots \, (a_r )_n } \over {(b_1 )_n \cdots \, (b_s )_n }}} \cr & { \times _r F_s \left[ {(a_1 \, + \,n;p,q;\kappa, \mu ),a_2 \, + \,n, \, \cdots \,, a_r \, + \,n \, b_1 \, + \,n,b_2 \, + \,n, \cdots, b_s \, + \,n \, ;z} \right]} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\, {(\Re (p) > 0, \, \,\Re (q) > 0,\, \, \Re (\kappa ) > 0, \, \,\Re (\mu ) > 0; \, \,n \in {\mathbb{N}}_0 ).}}
Proof
Differentiating (9) with respect to z and then replacing n → n + 1 in the right-hand side term, we obtain
(20) ![]()
d dz { r F s [ ( a 1 ; p , q ; κ , μ ) , a 2 , ⋯ , a r b 1 , b 2 , ⋯ , b s ; z ] } : = ∑ n = 0 ∞ ( a 1 ; p , q ; κ , μ ) n + 1 ( a 2 ) n + 1 ⋯ ( a r ) n + 1 ( b 1 ) n + 1 ( b 2 ) n + 1 ⋯ ( b s ) n + 1 z n + 1 ( n + 1 ) = a 1 ⋯ a r b 1 ⋯ b s r F s [ ( a 1 + 1 ; p , q ; κ , μ ) , a 2 + 1 , ⋯ , a r + 1 b 1 + 1 , b 2 + 1 , ⋯ , b s + 1 ; z ] , \eqalign{ & {d \over {dz}}\left\{ {{}_rF_s \left[ {(a_1 ;p,q;\kappa, \mu ),a_2, \, \cdots \,, a_r \, b_1, b_2, \cdots, b_s \, ;z} \right]} \right\}\,: = \,\sum\limits_{n = 0}^\infty {{(a_1 ;p,q;\kappa, \mu )_{n + 1} (a_2 )_{n + 1} \cdots (a_r )_{n + 1} } \over {(b_1 )_{n + 1} (b_2 )_{n + 1} \cdots (b_s )_{n + 1} }}{{z^{n + 1} } \over {(n + 1)}} \cr & = {{a_1 \cdots a_r } \over {b_1 \cdots b_s }} \, {}_rF_s \left[ {(a_1 + 1;p,q;\kappa, \mu ),a_2 + 1, \, \cdots \,, a_r + 1 \, b_1 + 1,b_2 + 1, \cdots, b_s + 1 \, ;z} \right],}
repeating the same procedure n -times gives the formula (19) .
Choosing r = s = 1 and r = 2, s = 1 in (19) , we have the derivative formulas for the (10) and (11) , respectively.
Corollary 8
The following derivative formulas hold true: (21) ![]()
d n dz n { Φ p , q κ , μ ( a ; b ; z ) } = ( a ) n ( b ) n Φ p , q κ , μ ( a + n ; b + n ; z ) {{d^n } \over {dz^n }} \, \{ \Phi _{p,q}^{\kappa, \mu } (a;b;z)\} = {{(a)_n } \over {(b)_n }} \, \Phi _{p,q}^{\kappa, \mu } (a + n;b + n;z) and (22) ![]()
d n dz n { F p , q κ , μ ( a , b , c ; z ) } = ( a ) n ( b ) n ( c ) n F p , q κ , μ ( a + n , b + n , c + n ; z ) . {{d^n } \over {dz^n }} \, \{ F_{p,q}^{\kappa, \mu } (a,b,c;z)\} = {{(a)_n (b)_n } \over {(c)_n }} \, F_{p,q}^{\kappa, \mu } (a + n,b + n,c + n;z).
The Bessel function J ν (z ) and the modified Bessel function I ν (z ) are expressible as hypergeometric functions as follows [11 , 12 , 33 ]:
(23) ![]()
J ν ( z ) = ( z 2 ) ν Γ ( ν + 1 ) 0 F 1 ( − ; ν + 1 ; − 1 4 z 2 ) ( ν ∈ ℂ \ ℤ − ( ℤ − = { − 1 , − 2 , − 3 , ... } ) ) \matrix{ {J_\nu (z) = {{({z \over 2})^\nu } \over {\Gamma (\nu + 1)}} \, _0 F_1 ( - ;\nu + 1; - {1 \over 4}z^2 )} \hfill \cr {(\nu \in {\mathbb{C}} \, \backslash \, {\mathbb{Z}}^ - ({\mathbb{Z}}^ - = \{ - 1, - 2, - 3,...\} ))}}
and
(24) ![]()
I ν ( z ) = ( z 2 ) ν Γ ( ν + 1 ) 0 F 1 ( − ; ν + 1 ; 1 4 z 2 ) ( ν ∈ ℂ \ ℤ − ) . \eqalign{ & {I_\nu (z) = {{({z \over 2})^\nu } \over {\Gamma (\nu + 1)}} \, _0 F_1 ( - ;\nu + 1;{1 \over 4}z^2 )} \hfill \cr & {(\nu \in {\mathbb{C}} \, \backslash \, {\mathbb{Z}}^ - ).}}
Additionally, for the incomplete gamma function γ (s,x ) defined by [10 ],
(25) ![]()
γ ( s , x ) = ∫ 0 x t s − 1 exp ( − t ) dt ( ℝ ( s ) > 0 ; x ≥ 0 ) . \eqalign{ & {\gamma (s,x) = \int\limits_0^x t^{s - 1} \exp ( - t)dt} \hfill \cr & {({\mathbb{R}}(s) > 0;x \ge 0).}}
Also, we know that [7 , 9 , 10 ]
(26) ![]()
1 F 1 [ s ; s + 1 ; − x ] = sx − s γ ( s , x ) . _1 F_1 [s;s + 1; - x] = sx^{ - s} \gamma (s,x).
So, we can deduce Corollary 9 and Corollary 10 by performing the relationships (23) –(26) in the equations (16) and (17) .
Corollary 9
Each of the following integral representations hold true: (27) ![]()
Φ p , q κ , μ ( a ; b + 1 ; − z ) = Γ ( b + 1 ) Γ ( a ) z − b 2 ∫ 0 ∞ t a − b 2 − 1 exp ( − t κ p − q t μ ) J b ( 2 zt ) dt \Phi _{p,q}^{\kappa, \mu } (a;b + 1; - z) = {{\Gamma (b + 1)} \over {\Gamma (a)}}z^{ - {b \over 2}} \int\limits_0^\infty t^{a - {b \over 2} - 1} \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }})J_b (2\sqrt {zt} )dt and (28) ![]()
Φ p , q κ , μ ( a ; b + 1 ; z ) = Γ ( b + 1 ) Γ ( a ) z − b 2 ∫ 0 ∞ t a − b 2 − 1 exp ( − t κ p − q t μ ) I b ( 2 zt ) dt \Phi _{p,q}^{\kappa, \mu } (a;b + 1;z) = {{\Gamma (b + 1)} \over {\Gamma (a)}}z^{ - {b \over 2}} \int\limits_0^\infty t^{a - {b \over 2} - 1} \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }})I_b (2\sqrt {zt} )dt on condition that the integrals involves are convergent.
Corollary 10
The following integral representation holds true: (29) ![]()
F p , q κ , μ ( a , b , b + 1 ; − z ) = bz − b Γ ( a ) ∫ 0 ∞ t a − b − 1 exp ( − t κ p − q t μ ) γ ( b , zt ) dt , F_{p,q}^{\kappa, \mu } (a,b,b + 1; - z) = {{bz^{ - b} } \over {\Gamma (a)}}\int\limits_0^\infty t^{a - b - 1} \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }})\gamma (b,zt)dt, on condition that the integrals involves are convergent.
4 A New Generalization of the extended Appell hypergeometric functions
In this section, we introduce extended Appell hypergeometric series and some extended multivariable hypergeometric functions.
Let us introduce the extensions of the Appell’s functions and extended Lauricella’s hypergeometric function and other functions defined by
(30) ![]()
p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] : = ∑ m , n = 0 ∞ ( a ; p , q ; κ , μ ) m + n ( b ) m ( c ) n ( d ) m + n x m m ! y n n ! max ( | x | , | y | ) < 1 , \matrix{ {{}_{p,q}F_1^{(\kappa, \mu )} [a,b,c;d;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m \, (c)_n } \over {(d)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\max (|x|, \, |y|) < \, 1,}} (31) ![]()
p , q F 2 ( κ , μ ) [ a , b , c ; d , e ; x , y ] : = ∑ m , n = 0 ∞ ( a ; p , q ; κ , μ ) m + n ( b ) m ( c ) n ( d ) m ( e ) n x m m ! y n n ! | x | + | y | < 1 , \matrix{ {_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m \, (c)_n } \over {(d)_m \, (e)_n }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {|x| + |y| < \, 1,}} (32) ![]()
p , q F 3 ( κ , μ ) [ a , b , c , d ; e ; x , y ] : = ∑ m , n = 0 ∞ ( a ; p , q ; κ , μ ) m ( b ) n ( c ) m ( d ) n ( e ) m + n x m m ! y n n ! max ( | x | , | y | ) < 1 , \matrix{ {_{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_m \, (b)_n \, (c)_m \, (d)_n } \over {(e)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\max (|x|, \, |y|) < \, 1,}} (33) ![]()
p , q F 4 ( κ , μ ) [ a , b ; c , d ; x , y ] : = ∑ m , n = 0 ∞ ( a ; p , q ; κ , μ ) m + n ( b ) m + n ( c ) m ( d ) n x m m ! y n n ! | x | + | y | < 1 , \matrix{ {{}_{p,q}F_4^{(\kappa, \mu )} [a,b;c,d;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_{m + n} } \over {(c)_m \, (d)_n }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\sqrt {|x|} + \sqrt {|y|} < \, 1,}} (34) ![]()
p , q F D ( κ , μ ; 3 ) [ a , b , c , d ; e ; x , y , z ] : = ∑ m , n , r = 0 ∞ ( a ; p , q ; κ , μ ) m + n + r ( b ) m ( c ) n ( d ) r ( e ) m + n + r x m m ! y n n ! z r r ! max ( | x | , | y | , | z | ) < 1 , \matrix{ {_{p,q} F_D^{(\kappa, \mu ;3)} [a,b,c,d;e;x,y,z]: = \, \sum\limits_{m,n,r = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n + r} \, (b)_m \, (c)_n \, (d)_r } \over { \, (e)_{m + n + r} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}} \, {{z^r } \over {r!}}} \cr {\max (|x|, \, |y|, \, |z|) < \, 1,}} (35) ![]()
p , q Φ 1 ( κ , μ ) [ a , b ; d ; x , y ] : = ∑ m , n = 0 ∞ ( a ; p , q ; κ , μ ) m + n ( b ) m ( d ) m + n x m m ! y n n ! max ( | x | , | y | ) < 1 , \matrix{ {_{p,q} \Phi _1^{(\kappa, \mu )} [a,b;d;x,y]: = \, \sum\limits_{m,n = 0}^\infty \, {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m } \over {(d)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\max (|x|, \, |y|) < \, 1,}} (36) ![]()
p , q Ψ 1 ( κ , μ ) [ a , b ; d , e ; x , y ] : = ∑ m , n = 0 ∞ ( a ; p , q ; κ , μ ) m + n ( b ) m ( d ) m ( e ) n x m m ! y n n ! | x | + | y | < 1 , \matrix{ {_{p,q} \Psi _1^{(\kappa, \mu )} [a,b;d,e;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m } \over {(d)_m \, (e)_n }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {|x| + |y| < \, 1,}} (37) ![]()
p , q Ξ 1 ( κ , μ ) [ a , c , d ; e ; x , y ] : = ∑ m , n = 0 ∞ ( a ; p , q ; κ , μ ) m ( c ) m ( d ) n ( e ) m + n x m m ! y n n ! max ( | x | , | y | ) < 1 \matrix{ {_{p,q} \Xi _1^{(\kappa, \mu )} [a,c,d;e;x,y]: = \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_m \, (c)_m \, (d)_n } \over {(e)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \cr {\max (|x|, \, |y|) < \, 1}}
and
(38) ![]()
p , q Φ D ( κ , μ ; 3 ) [ a , b , c ; e ; x , y , z ] : = ∑ m , n , r = 0 ∞ ( a ; p , q ; κ , μ ) m + n + r ( b ) m ( c ) n ( e ) m + n + r x m m ! y n n ! z r r ! max ( | x | , | y | , | z | ) < 1 , \matrix{ {_{p,q} \Phi _D^{(\kappa, \mu ;3)} [a,b,c;e;x,y,z]: = \, \sum\limits_{m,n,r = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n + r} \, (b)_m \, (c)_n } \over { \, (e)_{m + n + r} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}} \, {{z^r } \over {r!}}} \cr {\max (|x|, \, |y|, \, |z|) < \, 1,}}
respectively. Note that taking p = 1, q = 0, κ = 0 and μ = 0 gives the original ones [1 , 7 , 8 , 9 , 10 , 18 , 22 , 23 , 26 , 29 , 30 , 32 , 33 ]. Now, we obtain the integral representations of the functions (30) –(34) .
Theorem 11
The following integral representations for (30) hold true: (39) ![]()
p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] = 1 Γ ( a ) ∫ 0 ∞ t a − 1 exp ( − t κ p − q t μ ) Φ 2 [ b , c ; d ; xt , yt ] dt _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] = \, {1 \over {\Gamma (a)}}\int_0^\infty \, t^{a - 1} \, \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }}) \, \Phi _2 [b,c;d;xt,yt]dt and (40) ![]()
p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] = 1 Γ ( c ) ∫ 0 ∞ t c − 1 exp ( − t ) p , q Φ 1 ( κ , μ ) [ a , b ; d ; x , yt ] dt . _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] = \, {1 \over {\Gamma (c)}}\int_0^\infty \, t^{c - 1} \, \exp ( - t) \, _{p,q} \Phi _1^{(\kappa, \mu )} [a,b;d;x,yt]dt.
Proof
Using the generalization of the extended Pochhammer symbol (a 1 ; p,q ;κ,μ ) in the definition (30) by its integral representation given by (6) , we led to desired result (39) . Similar way, we can prove the (40) .
Theorem 12
The following integral representations for (31) hold true: (41) ![]()
p , q F 2 ( κ , μ ) [ a , b , c ; d , e ; x , y ] = 1 Γ ( a ) ∫ 0 ∞ t a − 1 exp ( − t κ p − q t μ ) 1 F 1 [ b ; d ; xt ] 1 F 1 [ c ; e ; yt ] dt _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] = \, {1 \over {\Gamma (a)}}\int_0^\infty \, t^{a - 1} \, \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }}) \, _1 F_1 [b;d;xt] \, _1 F_1 [c;e;yt]dt and (42) ![]()
p , q F 2 ( κ , μ ) [ a , b , c ; d , e ; x , y ] = 1 Γ ( c ) ∫ 0 ∞ t c − 1 exp ( − t ) p , q Ψ 1 ( κ , μ ) [ a , b ; d , e ; x , yt ] dt . _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] = \, {1 \over {\Gamma (c)}}\int_0^\infty \, t^{c - 1} \, \exp ( - t) \, _{p,q} \Psi _1^{(\kappa, \mu )} [a,b;d,e;x,yt]dt.
Proof
Using the generalization of the extended Pochhammer symbol (a 1 ; p,q ;κ, μ ) in the definition (31) by its integral representation given by (6) , we led to desired result (41) . Similar way, we can prove the (42) .
Theorem 13
The following integral representation for (32) holds true: (43) ![]()
p , q F 3 ( κ , μ ) [ a , b , c , d ; e ; x , y ] = 1 Γ ( b ) ∫ 0 ∞ t b − 1 exp ( − t ) p , q Ξ 1 ( κ , μ ) [ a , c , d ; e ; x , yt ] dt . _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y] = \, {1 \over {\Gamma (b)}}\int_0^\infty \, t^{b - 1} \, \exp ( - t) \, _{p,q} \Xi _1^{(\kappa, \mu )} [a,c,d;e;x,yt]dt.
Proof
Using the generalization of the extended Pochhammer symbol (a 1 ; p,q ;κ, μ ) in the definition (32) by its integral representation given by (3) , we led to desired result (43) .
Theorem 14
The following integral representation for (33) holds true: (44) ![]()
p , q F 4 ( κ , μ ) [ a , b ; c , d ; x , y ] = 1 Γ ( a ) ∫ 0 ∞ t a − 1 exp ( − t κ p − q t μ ) Ψ 2 [ b ; c , d ; xt , yt ] dt . _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y] = \, {1 \over {\Gamma (a)}}\int_0^\infty \, t^{a - 1} \, \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }}) \, \Psi _2 [b;c,d;xt,yt]dt.
Proof
Using the generalization of the extended Pochhammer symbol (a 1 ; p,q ;κ, μ ) in the definition (33) by its integral representation given by (6) , we led to desired result (44) .
Theorem 15
The following integral representations for (34) hold true: (45) ![]()
p , q F D ( κ , μ ; 3 ) [ a , b , c , d ; e ; x , y , z ] = 1 Γ ( a ) ∫ 0 ∞ t a − 1 exp ( − t κ p − q t μ ) Φ 2 ( 3 ) [ b , c , d ; e ; xt , yt , zt ] dt _{p,q} F_D^{(\kappa, \mu ;3)} [a,b,c,d;e;x,y,z] = \, {1 \over {\Gamma (a)}}\int_0^\infty \, t^{a - 1} \, \exp ( - {{t^\kappa } \over p} - {q \over {t^\mu }}) \, \Phi _2^{(3)} [b,c,d;e;xt,yt,zt]dt and (46) ![]()
p , q F D ( κ , μ ; 3 ) [ a , b , c , d ; e ; x , y , z ] = 1 Γ ( d ) ∫ 0 ∞ t d − 1 exp ( − t ) p , q Φ D ( κ , μ ; 3 ) [ a , b , c ; e ; x , y , zt ] dt . _{p,q} F_D^{(\kappa, \mu ;3)} [a,b,c,d;e;x,y,z] = \, {1 \over {\Gamma (d)}}\int_0^\infty \, t^{d - 1} \, \exp ( - t) \, _{p,q} \Phi _D^{(\kappa, \mu ;3)} [a,b,c;e;x,y,zt]dt.
Proof
Using the generalization of the extended Pochhammer symbol (a 1 ; p,q ;κ, μ ) in the definition (34) by its integral representation given by (6) , we led to desired result (45) . Similar way, we can prove the (46) .
Theorem 16
The following derivative formulas for (30) –(34) hold true: (47) ![]()
D x , y m , n { p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] } : = ( a ) m + n ( b ) m ( c ) n ( d ) m + n p , q F 1 ( κ , μ ) [ a + m + n , b + m , c + n ; d + m + n ; x , y ] , D_{x,y}^{m,n} \left\{ { \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}: = {{(a)_{m + n} (b)_m \, (c)_n } \over {(d)_{m + n} }}_{p,q} F_1^{(\kappa, \mu )} [a + m + n,b + m,c + n;d + m + n;x,y], (48) ![]()
D x , y m , n { p , q F 2 ( κ , μ ) [ a , b , c ; d , e ; x , y ] } : = ( a ) m + n ( b ) m ( c ) n ( d ) m ( e ) n p , q F 2 ( κ , μ ) [ a + m + n , b + m , c + n ; d + m , e + n ; x , y ] , D_{x,y}^{m,n} \left\{ { \, _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y]} \right\}: = {{(a)_{m + n} (b)_m \, (c)_n } \over {(d)_m \, (e)_n }}_{p,q} F_2^{(\kappa, \mu )} [a + m + n,b + m,c + n;d + m,e + n;x,y], (49) ![]()
D x , y m , n { p , q F 3 ( κ , μ ) [ a , b , c , d ; e ; x , y ] } : = ( a ) m ( b ) n ( c ) m ( d ) n ( e ) m + n p , q F 3 ( κ , μ ) [ a + m , b + n , c + m , d + n ; e + m + n ; x , y ] , D_{x,y}^{m,n} \left\{ { \, _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y]} \right\}: = {{(a)_m (b)_n \, (c)_m \, (d)_n } \over {(e)_{m + n} }}_{p,q} F_3^{(\kappa, \mu )} [a + m,b + n,c + m,d + n;e + m + n;x,y], (50) ![]()
D x , y m , n { p , q F 4 ( κ , μ ) [ a , b ; c , d ; x , y ] } : = ( a ) m + n ( b ) m + n ( c ) m ( c ) n p , q F 4 ( κ , μ ) [ a + m + n , b + m + n ; c + m , d + n ; x , y ] D_{x,y}^{m,n} \left\{ { \, _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y]} \right\}: = {{(a)_{m + n} (b)_{m + n} } \over {(c)_m \, (c)_n }}_{p,q} F_4^{(\kappa, \mu )} [a + m + n,b + m + n;c + m,d + n;x,y] and (51) ![]()
D x , y , z m , n , r { p , q F D ( κ , μ ; 3 ) [ a , b , c , d ; e ; x , y ] } : = ( a ) m + n + r ( b ) m ( c ) n ( d ) r ( e ) m + n + r × p , q F D ( κ , μ ; 3 ) [ a + m + n + r , b + m , c + n , d + r ; e + m + n + r ; x , y , z ] . \eqalign{ & D_{x,y,z}^{m,n,r} \{ \, _{p,q} F_D^{(\kappa, \mu ;3)} [a,b,c,d;e;x,y]\} : = {{(a)_{m + n + r} (b)_m \, (c)_n \, (d)_r } \over {(e)_{m + n + r} }} \cr & \times \, _{p,q} F_D^{(\kappa, \mu ;3)} [a + m + n + r,b + m,c + n,d + r;e + m + n + r;x,y,z].}
Proof
Differentiating (30) –(33) with respect to x and y , then repeating same procedure n -times and making some simple calculation, we can obtain the (47) –(50) results. Similiarly, taking differentiation (34) with respect to x , y and z , we can get the derivative formula (51)
Theorem 17
The following derivative formulas for (30) hold true: (52) ![]()
D y n { y c + n − 1 p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] } : = ( c ) n y p , q c − 1 F 1 ( κ , μ ) [ a , b , c + n ; d ; x , y ] , D_y^n \left\{ {y^{c + n - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}: = (c)_n \, y_{p,q}^{c - 1} F_1^{(\kappa, \mu )} [a,b,c + n;d;x,y], (53) ![]()
D y n { y d − 1 p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] } : = ( − 1 ) n ( 1 − d ) n y d − n − 1 p , q F 1 ( κ , μ ) [ a , b , c ; d − n ; x , y ] D_y^n \left\{ {y^{d - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}: = ( - 1)^n (1 - d)_n \, y^{d - n - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d - n;x,y] and (54) ![]()
D y n { y d − b − 1 p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] } : = ( − 1 ) n ( b − d − 1 ) n y d − b − n − 1 × ∑ n = 0 m ( m n ) ( a ) n ( c ) n y n ( d ) n ( d − b − m ) n p , q F 1 ( κ , μ ) [ a + n , b , c + n ; d + n ; x , y ] . \matrix{ {D_y^n \left\{ {y^{d - b - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}: = ( - 1)^n (b - d - 1)_n \, y^{d - b - n - 1} } \cr {\, \times \, \, \sum\limits_{n = 0}^m \left( {\matrix{ m \cr n \cr } } \right) \, {{(a)_n \, (c)_n \, y^n } \over {(d)_n \, (d - b - m)_n }} \, _{p,q} F_1^{(\kappa, \mu )} [a + n,b,c + n;d + n;x,y].}}
Proof
Multiplying the (30) with y c +n −1 and taking the derivative n -times with respect to y , we have
(55) ![]()
D y n { y c + n − 1 p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] } : = D y n { y c + n − 1 ∑ m , n = 0 ∞ ( a ; p , q ; κ , μ ) m + n ( b ) m ( c ) n ( d ) m + n x m m ! y n n ! } = ∑ m = 0 ∞ ( a ) m ( b ) m x m ( d ) m m ! D y n { y c + n − 1 ∑ n = 0 ∞ ( a + m ; p , q ; κ , μ ) n ( c ) n ( d + m ) n n ! y n } = ∑ m = 0 ∞ ( a ) m ( b ) m x m ( d ) m m ! ( c ) n y c − 1 ∑ n = 0 ∞ ( a + m ; p , q ; κ , μ ) n ( c + n ) n ( d + m ) n n ! y n = ( c ) n y c − 1 ∑ m , n = 0 ∞ ( a ; p , q ; κ , μ ) m + n ( b ) m ( c + n ) n ( d ) m + n x m m ! y n n ! . \matrix{ {D_y^n \left\{ {y^{c + n - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y]} \right\}:} \hfill & { = D_y^n \left\{ {y^{c + n - 1} \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m \, (c)_n } \over {(d)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}} \right\}} \hfill \cr {} \hfill & { = \sum\limits_{m = 0}^\infty {{(a)_m \, (b)_m \, x^m } \over {(d)_m \, m!}} \, D_y^n \left\{ {y^{c + n - 1} \, \sum\limits_{n = 0}^\infty {{(a + m;p,q;\kappa, \mu )_n \, (c)_n } \over {(d + m)_n \, n!}}y^n } \right\}} \hfill \cr {} \hfill & { = \sum\limits_{m = 0}^\infty {{(a)_m \, (b)_m \, x^m } \over {(d)_m \, m!}} \, (c)_n \, y^{c - 1} \, \sum\limits_{n = 0}^\infty {{(a + m;p,q;\kappa, \mu )_n \, (c + n)_n } \over {(d + m)_n \, n!}}y^n } \hfill \cr {} \hfill & { = \, (c)_n \, y^{c - 1} \, \sum\limits_{m,n = 0}^\infty {{(a;p,q;\kappa, \mu )_{m + n} \, (b)_m \, (c + n)_n } \over {(d)_{m + n} }} \, {{x^m } \over {m!}} \, {{y^n } \over {n!}}.}}
Thus, we obtain the (52) result. Similar way, we can prove the equations (53) and (54) .
Theorem 18
The following derivative formulas for (31) hold true: (56) ![]()
D y n { y c + n − 1 p , q F 2 ( κ , μ ) [ a , b , c ; d , e ; x , y ] } : = ( c ) n y p , q c − 1 F 2 ( κ , μ ) [ a , b , c + n ; d , e ; x , y ] D_y^n \left\{ {y^{c + n - 1} \, _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y]} \right\}: = (c)_n \, y_{p,q}^{c - 1} F_2^{(\kappa, \mu )} [a,b,c + n;d,e;x,y] and (57) ![]()
D y n { y e − 1 p , q F 2 ( κ , μ ) [ a , b , c ; d , e ; x , y ] } : = ( − 1 ) n ( 1 − e ) n y e − n − 1 p , q F 2 ( κ , μ ) [ a , b , c ; d , e − n ; x , y ] . D_y^n \left\{ {y^{e - 1} \, _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y]} \right\}: = ( - 1)^n (1 - e)_n \, y^{e - n - 1} \, _{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e - n;x,y].
Proof
The proof of theorem would be parallel to those of the Theorem 17 .
Theorem 19
The following derivative formulas for (32) hold true: (58) ![]()
D y n { y d + n − 1 p , q F 3 ( κ , μ ) [ a , b , c , d ; e ; x , y ] } : = ( d ) n y p , q d − 1 F 3 ( κ , μ ) [ a , b , c , d + n ; e ; x , y ] , D_y^n \left\{ {y^{d + n - 1} \, _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y]} \right\}: = (d)_n \, y_{p,q}^{d - 1} F_3^{(\kappa, \mu )} [a,b,c,d + n;e;x,y], (59) ![]()
D y n { ( 1 − y ) b + n − 1 p , q F 3 ( κ , μ ) [ a , b , e − c , c ; e ; x , y ] } : = ( − 1 ) n ( b ) n ( e − c ) n ( e ) n ( 1 − y ) b − 1 × p , q F 3 ( κ , μ ) [ a , b + n , e − c + n , d ; e + n ; x , y ] \matrix{ {D_y^n \left\{ {(1 - y)^{b + n - 1} \, _{p,q} F_3^{(\kappa, \mu )} [a,b,e - c,c;e;x,y]} \right\}: = ( - 1)^n {{(b)_n \, (e - c)_n } \over {(e)_n }} \, (1 - y)^{b - 1} } \cr { \times \, _{p,q} F_3^{(\kappa, \mu )} [a,b + n,e - c + n,d;e + n;x,y]}} and (60) ![]()
D y n { y e − c − 1 p , q F 3 ( κ , μ ) [ a , b , c , d ; e ; x , y ] } : = ( − 1 ) n ( c − e − 1 ) n y e − c − n − 1 × ∑ n = 0 m ( m n ) ( b ) n ( d ) n y n ( e ) n ( e − c − m ) n p , q F 3 ( κ , μ ) [ a , b + n , c , d + n ; e + n ; x , y ] . \matrix{ {D_y^n \left\{ {y^{e - c - 1} \, _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y]} \right\}: = ( - 1)^n (c - e - 1)_n \, y^{e - c - n - 1} } \cr { \times \, \sum\limits_{n = 0}^m \left( {\matrix{ m \cr n \cr } } \right) \, {{(b)_n \, (d)_n \, y^n } \over {(e)_n \, (e - c - m)_n }} \, _{p,q} F_3^{(\kappa, \mu )} [a,b + n,c,d + n;e + n;x,y].}}
Proof
The proof of theorem would be parallel to those of the Theorem 17 .
Theorem 20
The following derivative formulas for (33) hold true: (61) ![]()
D x n { x c − 1 p , q F 4 ( κ , μ ) [ a , b ; c , d ; x , y ] } : = ( − 1 ) n ( 1 − c ) n x p , q c − n − 1 F 4 ( κ , μ ) [ a , b ; c − n , d ; x , y ] D_x^n \left\{ {x^{c - 1} \, _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y]} \right\}: = ( - 1)^n (1 - c)_n \, x_{p,q}^{c - n - 1} F_4^{(\kappa, \mu )} [a,b;c - n,d;x,y] and (62) ![]()
D y n { y d − 1 p , q F 4 ( κ , μ ) [ a , b ; c , d ; x , y ] } : = ( − 1 ) n ( 1 − d ) n y p , q d − n − 1 F 4 ( κ , μ ) [ a , b ; c , d − n ; x , y ] . D_y^n \left\{ {y^{d - 1} \, _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y]} \right\}: = ( - 1)^n (1 - d)_n \, y_{p,q}^{d - n - 1} F_4^{(\kappa, \mu )} [a,b;c,d - n;x,y].
Proof
The proof of theorem would be parallel to those of the Theorem 17 .
5 Recursion Formulas for Extended Appell Hypergeometric Functions
In this section, we present some recursion formulas for Appell hypergeometric functions. Let’s we start following theorem.
Theorem 21
The following recursion formulas for (30) hold true: (63) ![]()
p , q F 1 ( κ , μ ) [ a , b + n , c ; d ; x , y ] = p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] + ax d ∑ k = 1 n p , q F 1 ( κ , μ ) [ a + 1 , b + k , c ; d + 1 ; x , y ] , _{p,q} F_1^{(\kappa, \mu )} [a,b + n,c;d;x,y] = _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] + \, {{ax} \over d} \, \sum\limits_{k = 1}^n \, _{p,q} F_1^{(\kappa, \mu )} [a + 1,b + k,c;d + 1;x,y], (64) ![]()
p , q F 1 ( κ , μ ) [ a , b − n , c ; d ; x , y ] = p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] − ax d ∑ k = 0 n − 1 p , q F 1 ( κ , μ ) [ a + 1 , b − k , c ; d + 1 ; x , y ] , _{p,q} F_1^{(\kappa, \mu )} [a,b - n,c;d;x,y] = _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] - \, {{ax} \over d} \, \sum\limits_{k = 0}^{n - 1} \, _{p,q} F_1^{(\kappa, \mu )} [a + 1,b - k,c;d + 1;x,y], and (65) ![]()
p , q F 1 ( κ , μ ) [ a , b , c ; d − n ; x , y ] = p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] + abx ∑ k = 1 n p , q F 1 ( κ , μ ) [ a + 1 , b + 1 , c ; d + 2 − k ; x , y ] ( d − k ) ( d − k − 1 ) + acy ∑ k = 1 n p , q F 1 ( κ , μ ) [ a + 1 , b , c + 1 ; d + 2 − k ; x , y ] ( d − k ) ( d − k − 1 ) . \eqalign{ & _{p,q} F_1^{(\kappa, \mu )} [a,b,c;d - n;x,y] = \,_{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \, abx \, \sum\limits_{k = 1}^n \, {{\,_{p,q} F_1^{(\kappa, \mu )} [a + 1,b + 1,c;d + 2 - k;x,y]} \over {(d - k)(d - k - 1)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \, acy \, \sum\limits_{k = 1}^n \, {{\,_{p,q} F_1^{(\kappa, \mu )} [a + 1,b,c + 1;d + 2 - k;x,y]} \over {(d - k)(d - k - 1)}}.}
Proof
Applying the transformation formula
( b + 1 ) m = ( b ) m × ( 1 + m b ) (b + 1)_m = (b)_m \times (1 + {m \over b})
in the definition of the extension of the Appell hypergeometric function
p , q F 1 ( κ , μ ) ( . ) \,_{p,q} F_1^{(\kappa, \mu )} (.)
in (30) and we have following contiguous formula:
(66) ![]()
p , q F 1 ( κ , μ ) [ a , b + 1 , c ; d ; x , y ] = p , q F 1 ( κ , μ ) [ a , b , c ; d ; x , y ] + ax d p , q F 1 ( κ , μ ) [ a + 1 , b + 1 , c ; d + 1 ; x , y ] . \,_{p,q}F_1^{(\kappa, \mu )} [a,b + 1,c;d;x,y] = \,_{p,q} F_1^{(\kappa, \mu )} [a,b,c;d;x,y] + \, {{ax} \over d} \, _{p,q} F_1^{(\kappa, \mu )} [a + 1,b + 1,c;d + 1;x,y].
Calculating the function
p , q F 1 ( κ , μ ) ( . ) \,_{p,q} F_1^{(\kappa, \mu )} (.)
with the parameter b +n by equation (66) for n times, we obtain the required result (63) . Setting the b = b − n in the equation (66) and making same calculation as above equation, we can be yield the desired result (64) . The proof of (65) is omitted to readers because it is similar to the proof of (63) .
Theorem 22
The following recursion formulas for (31) hold true: (67) ![]()
p , q F 2 ( κ , μ ) [ a , b + n , c ; d , e ; x , y ] = p , q F 2 ( κ , μ ) [ a , b , c ; d , e ; x , y ] + ax d ∑ k = 1 n p , q F 2 ( κ , μ ) [ a + 1 , b + k , c ; d + 1 , e ; x , y ] , \,_{p,q} F_2^{(\kappa, \mu )} [a,b + n,c;d,e;x,y] = \,_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] + \, {{ax} \over d} \, \sum\limits_{k = 1}^n \,_{p,q} F_2^{(\kappa, \mu )} [a + 1,b + k,c;d + 1,e;x,y], (68) ![]()
p , q F 2 ( κ , μ ) [ a , b − n , c ; d , e ; x , y ] = p , q F 2 ( κ , μ ) [ a , b , c ; d , e ; x , y ] − ax d ∑ k = 0 n − 1 p , q F 2 ( κ , μ ) [ a + 1 , b − k , c ; d + 1 , e ; x , y ] , \,_{p,q} F_2^{(\kappa, \mu )} [a,b - n,c;d,e;x,y] = \,_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] - \, {{ax} \over d} \, \sum\limits_{k = 0}^{n - 1} \,_{p,q} F_2^{(\kappa, \mu )} [a + 1,b - k,c;d + 1,e;x,y], and (69) ![]()
p , q F 2 ( κ , μ ) [ a , b , c ; d − n , e ; x , y ] = p , q F 2 ( κ , μ ) [ a , b , c ; d , e ; x , y ] + abx ∑ k = 1 n p , q F 2 ( κ , μ ) [ a + 1 , b + 1 , c ; d + 2 − k , e ; x , y ] ( d − k ) ( d − k − 1 ) . \,_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d - n,e;x,y] = \,_{p,q} F_2^{(\kappa, \mu )} [a,b,c;d,e;x,y] + \, abx \, \sum\limits_{k = 1}^n \, {{_{p,q} F_2^{(\kappa, \mu )} [a + 1,b + 1,c;d + 2 - k,e;x,y]} \over {(d - k)(d - k - 1)}}.
Proof
The proof of the Theorem 22 is similar to the proof of Theorem 21 .
Theorem 23
The following recursion formula for (32) holds true: (70) ![]()
p , q F 3 ( κ , μ ) [ a , b , c , d ; e − n ; x , y ] = p , q F 3 ( κ , μ ) [ a , b , c , d ; e ; x , y ] + acx ∑ k = 1 n p , q F 3 ( κ , μ ) [ a + 1 , b , c + 1 , d ; e + 2 − k ; x , y ] ( e − k ) ( e − k − 1 ) + bdy ∑ k = 1 n p , q F 3 ( κ , μ ) [ a , b + 1 , c , d + 1 ; e + 2 − k ; x , y ] ( e − k ) ( e − k − 1 ) . \eqalign{ & _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e - n;x,y] = _{p,q} F_3^{(\kappa, \mu )} [a,b,c,d;e;x,y] \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \, acx \, \sum\limits_{k = 1}^n \, {{_{p,q} F_3^{(\kappa, \mu )} [a + 1,b,c + 1,d;e + 2 - k;x,y]} \over {(e - k)(e - k - 1)}} \cr & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \, bdy \, \sum\limits_{k = 1}^n \, {{_{p,q} F_3^{(\kappa, \mu )} [a,b + 1,c,d + 1;e + 2 - k;x,y]} \over {(e - k)(e - k - 1)}}.}
Proof
The proof of the Theorem 23 is parallel to the proof of Theorem 21 .
Theorem 24
The following recursion formula for (33) holds true: (71) ![]()
p , q F 4 ( κ , μ ) [ a , b ; c − n , d ; x , y ] = p , q F 4 ( κ , μ ) [ a , b ; c , d ; x , y ] + abx ∑ k = 0 n − 1 p , q F 4 ( κ , μ ) [ a + 1 , b + 1 ; c + 1 − k , d ; x , y ] ( c − k ) ( c − k − 1 ) . \,_{p,q} F_4^{(\kappa, \mu )} [a,b;c - n,d;x,y] = \, _{p,q} F_4^{(\kappa, \mu )} [a,b;c,d;x,y] + \, abx \, \sum\limits_{k = 0}^{n - 1} \, {{\,_{p,q} F_4^{(\kappa, \mu )} [a + 1,b + 1;c + 1 - k,d;x,y]} \over {(c - k)(c - k - 1)}}.
Proof
The proof of the Theorem 24 is same as the proof of Theorem 21 .
Remark 2
Taking p = 1 and q = κ = μ = 0 in the relation Theorem 21 –Theorem 24 , it is easily seen that the special case of recursion formulas of Appell hypergeometric functions [32 ].