General series identities, some additive theorems on hypergeometric functions and their applications
aa r X i v : . [ m a t h . G M ] F e b GENERAL SERIES IDENTITIES, SOME ADDITIVE THEOREMS ONHYPERGEOMETRIC FUNCTIONS AND THEIR APPLICATIONS
MOHAMMAD IDRIS QURESHI, SAIMA JABEE AND MOHD SHADAB ∗ Abstract.
Motivated by the substantial development of the special functions, we con-tribute to establish some rigorous results on the general series identities with boundedsequences and hypergeometric functions with different arguments, which are generallyapplicable in nature. For the application purpose, we apply our results to some func-tions e.g. Trigonometric functions, Elliptic integrals, Dilogarithmic function, Errorfunction, Incomplete gamma function, and many other special functions. Introduction, Preliminaries and Notations
In present paper, we shall use the following standard notations: N := { , , , . . . } , N := { , , , , . . . } = N ∪ { } , Z − := { , − , − , − , . . . } , Z − := {− , − , − , . . . } = Z − \{ } , and Z = Z − ∪ N .Here, as usual, Z denotes the set of integers, R denotes the set of real numbers, R + denotes the set of positive real numbers and C denotes the set of complex numbers.The Pochhammer symbol (or the shifted factorial) ( λ ) ν ( λ, ν ∈ C ) is defined in terms ofthe familiar Gamma function, by( λ ) ν := Γ( λ + ν )Γ( λ ) = ν = 0; λ ∈ C \{ } ) λ ( λ + 1) . . . ( λ + n −
1) ; ( ν = n ∈ N ; λ ∈ C \ Z − ) ( − n k !( k − n )! ; ( λ = − k ; ν = n ; n, k ∈ N ; 0 ≤ n ≤ k )0 ; ( λ = − k ; ν = n ; n, k ∈ N ; n > k ) ( − n (1 − λ ) n ; ( ν = − n ; n ∈ N ; λ = 0 , ± , ± , . . . ) , (1.1)it being understood conventionally that (0) = 1, and assumed tacitly that the Gammaquotient exists.In the Gaussian hypergeometric series F ( a, b ; c ; z ), there are two numerator parameters a , b and one denominator parameter c . A natural generalization of this series is accom-plished by introducing any arbitrary number of numerator and denominator parameters. Mathematics Subject Classification.
Primary 33C20, 33EXX, 33BXX; Secondary 11B83.
Key words and phrases.
Fox-Wright hypergeometric function; Generalized hypergeometric function;Fifth roots of unity; Multiple bounded sequences.*Corresponding author.
The non-terminating hypergeometric series [9, p.42-43] p F q (cid:20) α , . . . , α p ; β , . . . , β q ; z (cid:21) = ∞ X n =0 ( α ) n . . . ( α p ) n ( β ) n . . . ( β q ) n z n n ! , (1.2)is known as the generalized Gauss and Kummer series , or simply, the generalized hyperge-ometric series . Here p and q are positive integers or zero (interpreting an empty productas unity), and we assume that the variable z , the numerator parameters α , . . . , α p andthe denominator parameters β , . . . , β q take on complex values, provided that β j = 0 , − , − , . . . ; j = 1 , . . . , q. (1.3)Convergence conditions [9, p.43] for generalized hypergeometric function are as follows:Suppose that none of the numerator parameters is zero or a negative integer (otherwisethe question of convergence will not arise), and with the usual restriction (1.3), the p F q series in the definition (1.2)(i) converges for | z | < ∞ , if p ≦ q ,(ii) converges for | z | <
1, if p = q + 1.Furthermore, if we denote ω = q X j =1 β j − p X j =1 α j , it is known that the p F q series, with p = q + 1, is(a) absolutely convergent for | z | = 1, if ℜ ( ω ) > | z | = 1, z = 1, if − < ℜ ( ω ) ≦ p Ψ q (cid:20) ( α , A ) , . . . , ( α p , A p );( β , B ) , . . . , ( β q , B q ); z (cid:21) = ∞ X n =0 Γ( α + A n ) . . . Γ( α p + A p n )Γ( β + B n ) . . . Γ( β q + B q n ) z n n != Γ( α ) . . . Γ( α p )Γ( β ) . . . Γ( β q ) ∞ X n =0 ( α ) nA . . . ( α p ) nA p ( β ) nB . . . ( β q ) nB q z n n ! , (1.4) p Ψ q (cid:20) ( α , A ) , . . . , ( α p , A p );( β , B ) , . . . , ( β q , B q ); z (cid:21) = Γ( α ) . . . Γ( α p )Γ( β ) . . . Γ( β q ) p Ψ ∗ q (cid:20) ( α , A ) , . . . , ( α p , A p );( β , B ) , . . . , ( β q , B q ); z (cid:21) , (1.5) p Ψ q (cid:20) ( α , A ) , . . . , ( α p , A p );( β , B ) , . . . , ( β q , B q ); z (cid:21) = 12 πρ Z L Γ( ζ ) p Y i =1 Γ( α i − A i ζ ) q Y j =1 Γ( β j − B j ζ ) ( − z ) − ζ dζ, (1.6)where ρ = − z ∈ C ; parameters α i , β j ∈ C ; coefficients A i , B j ∈ R = ( −∞ , + ∞ )in case of series (1.4) (or A i , B j ∈ R + = (0 , + ∞ ) in case of contour integral (1.6)), A i = 0 ( i = 1 , , ..., p ) , B j = 0 ( j = 1 , , ..., q ). In equation (1.4), the parameters α i , β j ENERAL SERIES IDENTITIES, SOME ADDITIVE THEOREMS ON HYPERGEOMETRIC FUNCTIONS3 and coefficients A i , B j are adjusted in such a way that the product of Gamma functionsin numerator and denominator should be well defined [1, 2].∆ ∗ = q X j =1 B j − p X i =1 A i , (1.7) δ ∗ = p Y i =1 | A i | − A i ! q Y j =1 | B j | B j , (1.8) µ ∗ = q X j =1 β j − p X i =1 α i + (cid:18) p − q (cid:19) , (1.9)and σ ∗ = (1 + A + ... + A p ) − ( B + ... + B q ) = 1 − ∆ ∗ . (1.10)Case(I): When contour (L) is a left loop beginning and ending at −∞ , then p Ψ q givenby (1.4) or (1.6) holds the following convergence conditionsi) When ∆ ∗ > − , < | z | < ∞ , z = 0 , ii) When ∆ ∗ = − , < | z | < δ ∗ , iii)When ∆ ∗ = − , | z | = δ ∗ , and ℜ ( µ ∗ ) > . Case(II): When contour (L) is a right loop beginning and ending at + ∞ , then p Ψ q givenby (1.4) or (1.6) holds the following convergence conditionsi) When ∆ ∗ < − , < | z | < ∞ , z = 0 , ii) When ∆ ∗ = − , | z | > δ ∗ , iii)When ∆ ∗ = − , | z | = δ ∗ , and ℜ ( µ ∗ ) > . Case(III): When contour (L) is starting from γ − i ∞ and ending at γ + i ∞ , where γ ∈ R = ( −∞ , + ∞ ), then p Ψ q is also convergent under the following conditionsi) When σ ∗ > , | arg( − z ) | < π σ ∗ , < | z | < ∞ , z = 0 , ii) When σ ∗ = 0 , arg( − z ) = 0 , < | z | < ∞ , z = 0 such that − γ ∆ ∗ + ℜ ( µ ∗ ) > + γ, iii)When γ = 0 , σ ∗ = 0 , arg( − z ) = 0 , < | z | < ∞ , z = 0 such that , ℜ ( µ ∗ ) > . Next we collect some results that we will need in the sequel.
Identity 1.
Let α = exp (cid:18) πi (cid:19) , i = q ( − r being non-negative integer, then1 + α r + α r + α r + α r = r ∈ { , , , , , . . . } r ∈ { , , , , , , , , , , , , . . . } . (1.11) Identity 2.
Let α = exp (cid:18) πi (cid:19) , i = q ( − M.I. QURESHI, S. JABEE AND M. SHADAB and r being non-negative integer, then1 + α r +1 + α r +2 + α r +3 + α r +4 = r ∈ { , , , , . . . } r ∈ { , , , , , , , , , , , , , , . . . } . (1.12)Above identities can be verified with the help of De Moivre’s theorem and some trigono-metrical identities. Gauss multiplication formula.
Let m being positive integer and n being non-negative integer, then ( b ) mn = m mn m Y j =1 (cid:16) b + j − m (cid:17) n . (1.13)Now, we are recalling some functions in the hypergeometric notations [7, pp. 71, 115](see also, [5]), which we will use in the applications. Table 1.
Some elementary functions and its hypergeometric representationsSer. No. Notation Hypergeometric Representation1 arcsin( x ) arcsin( x ) = x F , ; ; x x ) arctan( x ) = x F , ; ; − x x ) sin( x ) = x F − ; ; − x x )) (arcsin( x )) = x F , , , ; x x ) cos( x ) = F − ; ; − x (cid:18) √ (1 − x ) (cid:19) γ − (cid:18) √ (1 − x ) (cid:19) γ − = F γ, γ − ;2 γ ; x ENERAL SERIES IDENTITIES, SOME ADDITIVE THEOREMS ON HYPERGEOMETRIC FUNCTIONS5
Table 2.
Some special functions and its hypergeometric representationsSer. No. Notation Hypergeometric Representation1 Complete elliptic integral of first kind: K ( x ) K ( x ) = π F , ;1; x E ( x ) E ( x ) = π F − , ;1; x x ) erf( x ) = x √ ( π ) 1 F ; ; − x γ ( a, x ) γ ( a, x ) = x a a F a ;1 + a ; − x Li ( x ) Li ( x ) = x F , , , x General Series Identities
Theorem 1.
Suppose { φ ( r ) } ∞ r =0 is a bounded sequence of arbitrary real and complexnumbers and α = exp (cid:18) πi (cid:19) , i = q ( − then ∞ X r =0 φ ( r ) c r ( x ) r r ! + ∞ X r =0 φ ( r ) c r ( xα ) r r ! + ∞ X r =0 φ ( r ) c r ( xα ) r r ! + ∞ X r =0 φ ( r ) c r ( xα ) r r !+ ∞ X r =0 φ ( r ) c r ( xα ) r r ! = 5 ∞ X r =0 φ (5 r ) c r x r (5 r )! , (2.1) provided that each of the series involved is absolutely convergent.Proof. Suppose LHS of equation (2.1) is denoted by S , then S = ∞ X r =0 φ ( r ) c r x r r ! { α r + α r + α r + α r } . (2.2)Now, we apply Identity 1 in equation (2.2), we get S = 5 φ (0) c x (0)! + 5 φ (5) c x (5)! + 5 φ (10) c x (10)! + 5 φ (15) c x (15)! + . . . = 5 ∞ X r =0 φ (5 r ) c r x r (5 r )! . (2.3) (cid:3) M.I. QURESHI, S. JABEE AND M. SHADAB
Theorem 2.
Suppose { φ ( r ) } ∞ r =0 is a bounded sequence of arbitrary real and complexnumbers and α = exp (cid:18) πi (cid:19) , i = q ( − then ∞ X r =0 φ ( r ) c r ( x ) r r ! + α ∞ X r =0 φ ( r ) c r ( xα ) r r ! + α ∞ X r =0 φ ( r ) c r ( xα ) r r ! + α ∞ X r =0 φ ( r ) c r ( xα ) r r !+ α ∞ X r =0 φ ( r ) c r ( xα ) r r ! = 5 ∞ X r =0 φ (5 r + 2) c (5 r +2) x (10 r +4) (5 r + 2)! , (2.4) provided that each of the series involved is absolutely convergent.Proof. Suppose LHS of equation (2.4) is denoted by T , then T = ∞ X r =0 φ ( r ) c r x r r ! { α r +1 + α r +2 + α r +3 + α r +4 } . (2.5)Now, we apply Identity 2 in equation (2.5), we get T = 5 φ (2) c x (2)! + 5 φ (7) c x (7)! + 5 φ (12) c x (12)! + 5 φ (17) c x (17)! + . . . = 5 ∞ X r =1 φ (5 r − c (5 r − x (10 r − (5 r − . (2.6)Replacing r by r + 1 in equation (2.6), we get T = 5 ∞ X r =0 φ (5 r + 2) c (5 r +2) x (10 r +4) (5 r + 2)! . (2.7) (cid:3) Hypergeometric Representations
Any values of parameters and variables leading to the results, which do not makesense, are tacitly excluded.
Theorem 3.
Following sum of Fox-Wright hypergeometric functions with different ar-guments holds true: p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( x ) + p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) + p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) ENERAL SERIES IDENTITIES, SOME ADDITIVE THEOREMS ON HYPERGEOMETRIC FUNCTIONS7 + p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) + p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) = 5 Y i =1 Γ (cid:18) i (cid:19) p Ψ q +4 ( a , A ) , ( a , A ) , . . . , ( a p , A p ); (cid:16) , (cid:17) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , ( b , B ) , ( b , B ) , . . . , ( b q , B q ); cx ! , (3.1) where α = exp (cid:16) πi (cid:17) . Theorem 4.
Following sum of Fox-Wright hypergeometric functions with different ar-guments holds true: p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); cx +( α ) p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) +( α ) p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) +( α ) p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) +( α ) p Ψ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) = 5 c x Γ (cid:16) (cid:17) Γ (cid:16) (cid:17) Γ (cid:16) (cid:17) Γ (cid:16) (cid:17) × p Ψ q +4 ( a + 2 A , A ) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , ( b + 2 B , B ) , ( a + 2 A , A ) , . . . , ( a p + 2 A p , A p );( b + 2 B , B ) , . . . , ( b q + 2 B q , B q ); cx ! , (3.2) where α = exp (cid:16) πi (cid:17) . M.I. QURESHI, S. JABEE AND M. SHADAB
Proof.
On setting φ ( r ) = p Y j =1 Γ( a j + A j r ) q Y j =1 Γ( b j + B j r ) in general series identities (2.1) and (2.4), andapplying the definition of Fox-Wright hypergeometric function p Ψ q , we get equations(3.1), (3.2) respectively. (cid:3) Theorem 5.
Following sum of special case of Fox-Wright hypergeometric functions withdifferent arguments holds true: p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); cx + p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) + p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) + p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) + p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) = 5 p Ψ ∗ q +4 ( a , A ) , ( a , A ) , . . . , ( a p , A p ); (cid:16) , (cid:17) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , ( b , B ) , ( b , B ) , . . . , ( b q , B q ); cx ! , (3.3) where α = exp (cid:16) πi (cid:17) . Theorem 6.
Following sum of special case of Fox-Wright hypergeometric functions withdifferent arguments holds true: p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); cx +( α ) p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) ENERAL SERIES IDENTITIES, SOME ADDITIVE THEOREMS ON HYPERGEOMETRIC FUNCTIONS9 +( α ) p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) +( α ) p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) +( α ) p Ψ ∗ q ( a , A ) , ( a , A ) , . . . , ( a p , A p );( b , B ) , ( b , B ) , . . . , ( b q , B q ); c ( xα ) = ( a ) A . . . ( a p ) A p ( b ) B . . . ( b q ) B q c x × p Ψ ∗ q +4 ( a + 2 A , A ) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , (cid:16) , (cid:17) , ( b + 2 B , B ) , ( a + 2 A , A ) , . . . , ( a p + 2 A p , A p );( b + 2 B , B ) , . . . , ( b q + 2 B q , B q ); cx ! , (3.4) where α = exp (cid:16) πi (cid:17) . Proof.
On setting φ ( r ) = ( a ) rA ( a ) rA ... ( a p ) rAp ( b ) rB ( b ) rB ... ( b q ) rBq in general series identities (2.1) and (2.4),and applying the definition of Fox-Wright hypergeometric function p Ψ ∗ q , we get equations(3.3), (3.4) respectively. (cid:3) Theorem 7.
Following sum of generalized hypergeometric functions with different ar-guments holds true: p F q ( a p );( b q ); cx + p F q ( a p );( b q ); c ( xα ) + p F q ( a p );( b q ); c ( xα ) + p F q ( a p );( b q ); c ( xα ) + p F q ( a p );( b q ); c ( xα ) = 5 p F q +4 ( a p )5 , a p )5 , a p )5 , a p )5 , a p )5 ; , , , , ( b q )5 , b q )5 , b q )5 , b q )5 , b q )5 ; cx (1+ q − p ) ! , (3.5) where α = exp (cid:16) πi (cid:17) ; 5 p ≤ q + 4 , (cid:12)(cid:12)(cid:12) (cid:16) cx (1+ q − p ) (cid:17) (cid:12)(cid:12)(cid:12) < ∞ ; p = q + 1 , (cid:12)(cid:12)(cid:12) (cid:16) cx (1+ q − p ) (cid:17) (cid:12)(cid:12)(cid:12) < . Theorem 8.
Following sum of generalized hypergeometric functions with different ar-guments holds true: p F q ( a p );( b q ); cx + ( α ) p F q ( a p );( b q ); c ( xα ) + ( α ) p F q ( a p );( b q ); c ( xα ) +( α ) p F q ( a p );( b q ); c ( xα ) + ( α ) p F q ( a p );( b q ); c ( xα ) = 5 c x Q pi =1 ( a i ) Q qi =1 ( b i ) p F q +4 a p )5 , a p )5 , a p )5 , a p )5 , a p )5 ; , , , , b q )5 , b q )5 , b q )5 , b q )5 , b q )5 ; cx (1+ q − p ) ! , (3.6) where α = exp (cid:16) πi (cid:17) ; 5 p ≤ q + 4 , (cid:12)(cid:12)(cid:12) (cid:16) cx (1+ q − p ) (cid:17) (cid:12)(cid:12)(cid:12) < ∞ ; p = q + 1 , (cid:12)(cid:12)(cid:12) (cid:16) cx (1+ q − p ) (cid:17) (cid:12)(cid:12)(cid:12) < . Proof.
On setting φ ( r ) = ( a ) r ( a ) r ... ( a p ) r ( b ) r ( b ) r ... ( b q ) r in general series identities (2.1) and (2.4), andapplying the definition of generalized hypergeometric function p F q , we get equations(3.5), (3.6) respectively. (cid:3) Applications
As the direct application of our theorems 7 & 8, we obtain following results on the sumof special functions and elementary functions with different arguments for α = exp (cid:16) πi (cid:17) :On setting p = 0 , q = 1 , c = − and b = in equation (3.5), we obtain α sin x + α sin( xα ) + α sin( xα ) + α sin( xα ) + sin( xα )= 5 xα F ————————– ; , , , , , , , , ; − (cid:18) x (cid:19) . (4.1)On setting p = 0 , q = 1 , c = − and b = in equation (3.6), we obtainsin x + sin( xα ) + sin( xα ) + sin( xα ) + sin( xα )= x F ————————– ; , , , , , , , , ; − (cid:18) x (cid:19) . (4.2)On setting p = 0 , q = 1 , c = − and b = in equation (3.5), we obtaincos x + cos( xα ) + cos( xα ) + cos( xα ) + cos( xα )= 5 F ————————– ; , , , , , , , , ; − (cid:18) x (cid:19) . (4.3) ENERAL SERIES IDENTITIES, SOME ADDITIVE THEOREMS ON HYPERGEOMETRIC FUNCTIONS11
On setting p = 0 , q = 1 , c = − and b = in equation (3.6), we obtaincos x + ( α ) cos( xα ) + ( α ) cos( xα ) + ( α ) cos( α ) + ( α ) cos( α )= 5 x F ————————– ; , , , , , , , , ; − (cid:18) x (cid:19) . (4.4)On setting p = 2 , q = 1 , c = − a = , a = 1 , b = in equation (3.5), we obtain α arctan x + α arctan( xα ) + α arctan( xα ) + α arctan( xα ) + arctan( xα )= 5 xα F , ; ; − x . (4.5)On setting p = 2 , q = 1 , c = − a = , a = 1 , b = in equation (3.6), we obtainarctan x + arctan( xα ) + arctan( xα ) + arctan( xα ) + arctan( xα )= x F , ; ; − x = arctan( x ) . (4.6)On setting p = 2 , q = 1 , c = 1 and a = , a = − , b = 1 in equation (3.5), we obtain E ( x ) + E ( xα ) + E ( xα ) + E ( xα ) + E ( xα )= (cid:18) π (cid:19) F − , , , , , , , , , ; , , , , , , , , x . (4.7)On setting p = 2 , q = 1 , c = 1 and a = , a = − , b = 1 in equation (3.6), we obtain E ( x ) + α E ( xα ) + α E ( xα ) + α E ( xα ) + α E ( xα )= − x π ! F , , , , , , , , , ; , , , , , , , , x . (4.8)On setting p = 1 , q = 1 , c = − a = , b = in equation (3.5), we obtain α erf ( x ) + α erf ( xα ) + α erf ( xα ) + α erf ( xα ) + erf ( xα )= xα p ( π ) ! F ; , , , , ; − x ! . (4.9)On setting p = 1 , q = 1 , c = − a = , b = in equation (3.6), we obtainerf ( x ) + erf ( xα ) + erf ( xα ) + erf ( xα ) + erf ( xα )= x p ( π ) ! F ; , , , , ; − x ! . (4.10) On setting p = 3 , q = 2 , c = 1 and a = a = a = 1, b = 2 , b = in equation (3.5),we obtain α (arcsin x ) + α (arcsin( xα )) + α (arcsin( xα )) + α (arcsin( xα )) + (arcsin( xα )) = 5 x α F , , , , , , , , , , , ; x . (4.11)On setting p = 3 , q = 2 , c = 1 and a = a = a = 1, b = 2 , b = in equation (3.6), weobtain α (arcsin x ) + α (arcsin( xα )) + α (arcsin( xα )) + α (arcsin( xα )) + (arcsin( xα )) = 8 x α F , , , , , , , , , , , ; x . (4.12)On setting p = 2 , q = 1 , c = 1 and a = a = , b = 1 in equation (3.5), we obtain K ( x ) + K ( xα ) + K ( xα ) + K ( xα ) + K ( xα )= (cid:18) π (cid:19) F , , , , , , , , , ; , , , , , , , , x . (4.13)On setting p = 2 , q = 1 , c = 1 and a = a = , b = 1 in equation (3.6), we obtain K ( x ) + α K ( xα ) + α K ( xα ) + α K ( xα ) + α K ( xα )= x π ! F , , , , , , , , , ; , , , , , , , , x . (4.14)On setting p = 3 , q = 2 , c = 1 and a = a = a = 1 , b = b = 2 in equation (3.5), weobtain α Li ( x ) + α Li (( xα ) ) + α Li (( xα ) ) + α Li (( xα ) ) + Li (( xα ) )= (cid:16) x α (cid:17) F , , , ; x . (4.15)On setting p = 3 , q = 2 , c = 1 and a = a = a = 1 , b = b = 2 in equation (3.6), weobtain α Li ( x ) + α Li (( xα ) ) + α Li (( xα ) ) + αLi (( xα ) ) + Li (( xα ) )= x α ! F , , , ; x . (4.16) ENERAL SERIES IDENTITIES, SOME ADDITIVE THEOREMS ON HYPERGEOMETRIC FUNCTIONS13
On setting p = 1 , q = 1 , c = − a = a, b = 1 + a in equation (3.5), we obtain α a γ ( a, x ) + α a γ ( a, ( xα ) ) + α a γ ( a, ( xα ) ) + α a γ ( a, ( xα ) ) + γ ( a, ( xα ) )= x a α a a ! F a ; a +55 , , , , ; − x ! . (4.17)On setting p = 1 , q = 1 , c = − a = a, b = 1 + a in equation (3.6), we obtain α a γ ( a, x ) + α a +1 γ ( a, ( xα ) ) + α a +2 γ ( a, ( xα ) ) + α a +3 γ ( a, ( xα ) ) + α γ ( a, ( xα ) )= x a +4 α a a + 2) ! F a +25 ; a +75 , , , , ; − x ! . (4.18)On setting p = 2 , q = 1 , c = 1 and a = γ, a = γ − , b = 2 γ in equation (3.5), weobtain
21 + p (1 − ( x ) ) ! γ − +
21 + p (1 − ( xα ) ) ! γ − +
21 + p (1 − ( xα ) ) ! γ − +
21 + p (1 − ( xα ) ) ! γ − +
21 + p (1 − ( xα ) ) ! γ − = 5 F γ , γ +15 , γ +25 , γ +35 , γ +45 , γ − , γ +110 , γ +310 , γ +510 , γ +710 ; , , , , γ , γ +15 , γ +25 , γ +35 , γ +45 ; x . (4.19)On setting p = 2 , q = 1 , c = 1 and a = γ, a = γ − , b = 2 γ in equation (3.6), weobtain
21 + p (1 − ( x ) ) ! γ − + α
21 + p (1 − ( xα ) ) ! γ − + α
21 + p (1 − ( xα ) ) ! γ − + α
21 + p (1 − ( xα ) ) ! γ − + α
21 + p (1 − ( xα ) ) ! γ − = γ ) ( γ − ) x γ ) ! F γ +25 , γ +35 , γ +45 , γ +55 , γ +65 , γ +310 , γ +510 , γ +710 , γ +910 , γ +1110 ; , , , , γ +25 , γ +35 , γ +45 , γ +55 , γ +65 ; x . (4.20) Remark:
Making suitable adjustments of parameters and variables in Theorems3,4,5 and 6, we can derive some more results involving generalised Bessel functionsΦ( α, β ; z ) or J µν ( z ), Mittag-Leffler functions E α ( z ) and its generalizations E α,β ( z ). SinceWright’s generalized function p Ψ q of one variable is the particular case of Fox H -functionof one variable. Therefore, for more special cases of p Ψ q , we refer two monographs ofMathai-Saxena [6] and Srivastava, Gupta and Goyal [8]. Conclusion
Here, we have established some results on the sum of hypergeometric functions withdifferent arguments. We applied our results to Trigonometric functions, Elliptic inte-grals, Dilogarithmic function, Error function, and Incomplete gamma function.One can also establish the some results on the sum of these special functions for exam-ple: ordinary Bessel function J ν (x), modified Bessel function I ν (x), complete elliptic in-tegrals B (x), C (x), D (x), Lerch’s transcendent Φ( x, q, a ), Fresnel’s integrals S(x), S (x),S (x), C ∗ (x), C (x), C (x), Sine integral S i (x), hyperbolic sine integral Sh i (x), Polyloga-rithm function Li q (x), Sturve function H ν (x), Modified Sturve functions L ν (x), h µ,ν (x),Lommel function s µ,ν (x), Kelvin’s functions ber( x ) , bei(x), Incomplete beta functionB x ( α, β ), Hyperbessel function of Humbert J m,n (x), Modified hyperbessel function ofDelerue I m,n (x), Arctangents function Ti (x), sinh( x ) , cosh( x ) , tanh − ( x ) , sinh − ( x ), sinh − ( x ) √ (1+ x ) , (cid:16) sinh − x (cid:17) , sin (cid:0) a sin − x (cid:1) , cos (cid:0) a sin − x (cid:1) , log e (1 ± x ), arcsin( x ) √ (1 − x ) and exp ( a sin − x )etc. References [1] Boersma, J.; On a function which is a special case of Meijer’s G -function, Compositio Math. , (1962), 34-63.[2] Braaksma, B.L.J.; Asymptotic expansions and analytic continuations for a class of Barnes-integrals, Compositio Math. , (1964), 239-341.[3] Fox, C.; The asymptotic expansion of generalized hypergeometric functions, Proc. London Math.Soc. , (2) (1928), 389-400.[4] Fox, C.; The G and H functions as symmetrical Fourier kernels, Trans. Amer. Math. Soc. , (1961),395-421.[5] Gradshteyn, I.S. and Ryzhik, I.M.; Table of integrals, series and products , 8th ed., Academic PressInc., San Diego, CA. 2014.[6] Mathai, A.M. and Saxena, R.K.;
The H-function with applications in statistics and other disciplines ,John Wiley and Sons (Halsted Press), New York, 1978.[7] Rainville, E.D.;
Special Functions,
The Macmillan Co. Inc.,New York,1960;Reprinted by ChelseaPubl. Co. Bronx, New York, 1971.[8] Srivastava, H.M. Gupta, K.C. and Goyal, S.P.;
The H-functions of one and two variables withapplications,
South Asian Publishers, New Delhi and Madras, 1982.[9] Srivastava, H.M. and Manocha, H.L.;
A Treatise on Generating functions,
Halsted Press (Ellis Hor-wood Ltd., Chichester, U.K.), John Wiley and Sons, New York, Chichester, Brisbane and Toronto,1984.[10] Wright, E.M.; The asymptotic expansion of the generalized hypergeometric function- I,
J. LondonMath. Soc. , (4) (1935), 286-293.[11] Wright, E.M.; The asymptotic expansion of the generalized hypergeometric function-II, Proc. Lon-don Math. Soc.(2) , (1940), 389-408. Mohammad Idris Qureshi: Department of Applied Sciences and Humanities, Faculty ofEngineering and Technology, Jamia Millia Islamia (A Central University), New Delhi110025, India.
E-mail address : miqureshi [email protected] ENERAL SERIES IDENTITIES, SOME ADDITIVE THEOREMS ON HYPERGEOMETRIC FUNCTIONS15
Saima Jabee: Department of Applied Sciences and Humanities, Faculty of Engineeringand Technology, Jamia Millia Islamia (A Central University), New Delhi 110025, India.
E-mail address : [email protected] Mohammad Shadab: Department of Applied Sciences and Humanities, Faculty of Engi-neering and Technology, Jamia Millia Islamia (A Central University), New Delhi 110025,India.
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