Some decomposition formulas of generalized hypergeometric functions and formulas of an analytic continuation of the Clausen function
aa r X i v : . [ m a t h - ph ] O c t Some decomposition formulas of generalized hypergeometricfunctions and formulas of an analytic continuation of theClausen function
A. Hasanov
Institute of Mathematics, Uzbek Academy of Sciences,29, F. Hodjaev street, Tashkent 100125, UzbekistanE-mail: [email protected]
Abstract
In this paper, using similar symbolical method of Burchnall and Chaundy formulas of expansionfor the generalized hypergeometric function were constructed. By means of the found formulas ofexpansion the formulas of an analytic continuation for hypergeometric function of Clausen is defined.The obtained formulas of an analytic continuation express known hypergeometric Appell function F ( a ; b , b ; c , c ; x, y ) which theory is well studied. MSC: primary 33C20; secondary 44A45.Key Words and Phrases : Generalized hypergeometric series; Inverse pairs of symbolic; Decompo-sition formulas; Gauss function; Clausen function; formulas Analytic continuations.
Hardly there is a necessity to speak about importance of properties of hypergeometric functions for anyscientist and the engineer dealing with practical application of differential equations. The solution ofvarious problems concerning a thermal conduction and dynamics, electromagnetic oscillations and aero-dynamics, a quantum mechanics and the theory of potentials, leads to hypergeometric functions. Moreoften they appear at solving of partial differential equations by the method of a separation of variables.A variety of the problems leading hypergeometric functions, has called fast growth of number of thefunctions, applied in applications (for example, in the monography [3] 205 hypergeometric functions arestudied). There were monographies and papers on the theory of special functions. But in these mono-graphs there is no formula of expansion and an analytic continuation of the generalized hypergeometricfunction. In this paper, using similar symbolical method Burchnall and Chaundy, we shall constructformulas of expansion for the generalized hypergeometric function. By means of the obtained formulas ofexpansion we find the formulas of an analytic continuation of hypergeometric function of Clausen. Thefound formulas of an analytic continuation express known hypergeometric Appell function ([1], p. 14,(12), p.28, (2) and [2]) F ( a ; b , b ; c , c ; x, y ) = ∞ X i,j =0 ( a ) i + j ( b ) i ( b ) j ( b ) i ( b ) j i ! j ! x i y j , | x | + | y | < , ( a ; b , b ; c , c ; x, y ) = Γ ( c ) Γ ( c )Γ ( b ) Γ ( c − b ) Γ ( b ) Γ ( c − b ) ×× Z Z ξ b − η b − (1 − ξ ) c − b − (1 − η ) c − b − (1 − xξ − yη ) − a dξdη, Re c i > Re b i > , i = 1 , ., which theory is well studied.In the Gaussian [1, 2], hypergeometric series F ( a, b ; c ; x ) there are two numeration parameters a, b and one denominator parameter c . A natural generalization of this series is accomplished by introducingany arbitrary number of numerator and denominator parameters. The resulting series [1-3] p F q α , α , ..., α p ; β , β , ..., β q ; z = ∞ X m =0 ( α ) m ( α ) m ... ( α p ) m ( β ) m ( β ) m ... ( β q ) m m ! z m , (1 . λ ) n is the Pochhammer symbol defined( λ ) n = Γ ( λ + n )Γ ( λ ) , λ = 0 , − , − , ..., (1 . p and q are positive integers or zero (interpreting an empty product as 1), and we assume that the variable z ,the numerator parameters α , ..., α p , and the denominator parameters β , ..., β q take on complex values,provided that β i = 0 , − , − , ... ; j = 1 , ..., q. (1 . | z | < ∞ if p ≤ q ,(II) converges for | z | < ∞ if p = q + 1, and(III) diverges for all z , , if p > q + 1.Furthermore, if we set ω = q X i =1 β i − p X i =1 α i , (1 . p F q series, with p = q + 1, is(IV) absolutely converges for if Re ( ω ) > z = 1, if − < Re ( ω ) ≤
0, and(VI) divergent for | z | = 1 if Re ( ω ) ≤ − F ( a, b ; c ; x ) can be found in the monography [1, 2]. In the paper ([4], p. 704, (3)), for hypergeometricfunction of Clausen (1.1) in the case of p = 3 , q = 2 the formulas of an analytic continuation were foundby the author. One of the found formulas has a form of F a , a , a b , b x = F R a , a , a b , b x + ξ a , a , a b , b x , (1 . F R a , a , a b , b x = Γ ( b ) Γ ( b ) Γ ( b + b − a − a − a )Γ ( a ) Γ ( b + b − a − a ) Γ ( b + b − a − a ) × ∞ X n =0 ( b − a ) n ( b − a ) n ( b + b − a − a − a ) n ( b + b − a − a ) n ( b + b − a − a ) n n ! × F ( a , a ; a + a + a − b − b − n + 1; 1 − x ) , (1 . ξ a , a , a b , b x = Γ ( b ) Γ ( b ) Γ ( a + a + a − b − b )Γ ( a ) Γ ( a ) Γ ( a ) × x a − b − b +1 (1 − x ) b + b − a − a − a ∞ X n =0 ( b − a ) n ( b − a ) n ( b + b − a − a − a + 1) n n ! (cid:18) x − x (cid:19) n × F (1 − a , − a ; b + b − a − a − a + n + 1; 1 − x ) . (1 . F is very bulky and not convenient in applications, and does not express throughknown hypergeometric functions. Burchnall and Chaundy ([5] and [6]) and (Chaundy [7]) systematically presented a number of expansionand decomposition formulas for double hypergeometric functions in series of simpler hypergeometricfunctions. Their method is based upon the following inverse pairs of symbolic operators: ∇ x,y ( h ) := Γ ( h ) Γ ( δ + δ ′ + h )Γ ( δ + h ) Γ ( δ ′ + h ) = ∞ X i =0 ( − δ ) i ( − δ ′ ) i ( h ) i i ! , (2 . x,y ( h ) := Γ ( δ + h ) Γ ( δ ′ + h )Γ ( h ) Γ ( δ + δ ′ + h ) = ∞ X i =0 ( − δ ) i ( − δ ′ ) i (1 − h − δ − δ ′ ) i i != ∞ X i =0 ( − i ( h ) i ( − δ ) i ( − δ ′ ) i ( h + i − i ( δ + h ) i ( δ ′ + h ) i i ! , (cid:18) δ := x ∂∂x ; δ ′ := y ∂∂y (cid:19) . (2 . F (3) A , F E , F K , F M , F P and F T , H A , H C , respectively (see, for definitions, [3, Section 1.5] and [10], p. 66et seq.), by Singhal and Bhati [11], Hasanov and Srivastava [14, 15] for deriving analogous multiple-seriesexpansions associated with several multivariable hypergeometric functions. We now introduce here thefollowing analogues of Burchnall-Chaundy symbolic operators ∇ x,y and ∆ x,y defined by (2.1) and (2.2),respectively: H x ,...,x l ( α, β ) := Γ ( β ) Γ ( α + δ + · · · + δ l )Γ ( α ) Γ ( β + δ + · · · + δ l ) = ∞ X k , ··· ,k l =0 ( β − α ) k + ··· + k l ( − δ ) k · · · ( − δ l ) k l ( β ) k + ··· + k l k ! · · · k l ! (2 . H x ,...,x l ( α, β ) := Γ ( α ) Γ ( β + δ + · · · + δ l )Γ ( β ) Γ ( α + δ + · · · + δ l ) = ∞ X k ,...k l =0 ( β − α ) k + ··· + k l ( − δ ) k · · · ( − δ l ) k l (1 − α − δ − · · · − δ l ) k + ··· + k l k ! · · · k l ! (2 . δ j := x j ∂∂x j , j = 1 , ..., l ; l ∈ N := { , , , ... } (cid:19) . By means of operators (2.3) - (2.4) we shall construct functional identities for the generalized hypergeo-metric function (1.1).
Similarly just as in the papers [5, 6], the following functional identities p F q α , α , ..., α p ; β , β , ..., β p ; x = H x ( α p , β q ) p − F q − α , α , ..., α p − ; β , β , ..., β q − ; x , p ≥ , q ≥ , (3 . p − F q − α , α , ..., α p − ; β , β , ..., β q − ; x = ¯ H x ( α p , β q ) p F q α , α , ..., α p ; β , β , ..., β p ; x , p ≥ , q ≥ , (3 . p F q α , α , ..., α p ; β , β , ..., β q ; x = H x ( α p , β q ) H x ( α p − , β q − ) p − F q − α , α , ..., α p − ; β , β , ..., β q − ; x , p ≥ , q ≥ , (3 . p − F q − α , α , ..., α p − ; β , β , ..., β q − ; x = ¯ H x ( α p , β q ) ¯ H x ( α p − , β q − ) p F q α , α , ..., α p ; β , β , ..., β p ; x , p ≥ , q ≥ . Applying operators (2.3) - (2.3), from functional identities we define the following expansions for thegeneralized hypergeometric function (1.1) p F q α , α , ..., α p ; β , β , ..., β p ; x = ∞ X i =0 ( − i ( α ) i ( α ) i ... ( α p − ) i ( β q − α p ) i ( β ) i ( β ) i ... ( β q − ) i ( β q ) i i ! x ip − F q − α + i, α + i, ..., α p − + i ; β + i, β + i, ..., β q − + i ; x , p ≥ , q ≥ , (4 . p − F q − α , α , ..., α p − ; β , β , ..., β q − ; x = ∞ X i =0 ( α ) i ( α ) i ... ( α p − ) i ( β q − α p ) i ( β ) i ( β ) i ... ( β p ) i i ! x i p F q α + i, α + i, ..., α p − + i, α p ; β + i, β + i, ..., β q + i ; x , p ≥ , q ≥ , (4 . F q α , α , ..., α p ; β , β , ..., β q ; x = ∞ X i,j =0 ( − i + j ( α ) i + j ( α ) i + j ... ( α p − ) i + j ( α p − ) i ( β q − − α p − ) j ( β q − α p ) i ( β ) i + j ( β ) i + j ... ( β q − ) i + j ( β q − ) i + j ( β q ) i i ! j ! x i + j × p − F q − α + i, α + i, ..., α p − + i ; β + i, β + i, ..., β q − + i ; x , p ≥ , q ≥ , (4 . p − F q − α , α , ..., α p − ; β , β , ..., β q − ; x = ∞ X i,j =0 ( α ) i + j ( α ) i + j ... ( α p − ) i + j ( α p ) j ( β q − ) i ( β q − − α p − ) i + j ( β q − α p ) i ( β ) i + j ( β ) i + j ... ( β q ) i + j ( β q − − α p − ) i i ! j ! x i + j × p F q α + i + j, α + i + j, ..., α p − + i + j, α p − , α p + j ; β + i + j, β + i + j, ..., β q + i + j ; x , p ≥ , q ≥ , (4 . p F q α , α , ..., α p ; β , β , ..., β q ; x + y − xy = ∞ X i =0 ( − i ( α ) i ( α ) i ... ( α p ) i ( β ) i ( β ) i ... ( β q ) i i ! x i y ip F q α + i, α + i, ..., α p + i ; β + i, β + i, ..., β q + i ; x + y , (4 . p F q α , α , ..., α p ; β , β , ..., β q ; x + y = ∞ X i =0 ( α ) i ( α ) i ... ( α p ) i ( β ) i ( β ) i ... ( β q ) i i ! x i y ip F q α + i, α + i, ..., α p + i ; β + i, β + i, ..., β q + i ; x + y − xy . (4 . p F q α , α , ..., α p ; β , β , ..., β p ; x = ∞ X i =0 ( β q − α p ) i ( − δ ) i ( β q ) i i ! p − F q − α , α , ..., α p − ; β , β , ..., β q − ; x . (4 . f ( z ) takes place identities( − δ ) i f ( z ) = ( − δ ) (1 − δ ) · · · ( i − − δ ) f ( z ) = ( − i z i d i dz i f ( z ) . (4 . d i dx i p F q α , α , ..., α p ; β , β , ..., β p ; x = ( α ) i ( α ) i ... ( α p ) i ( β ) i ( β ) i ... ( β p ) i p F q α + i, α + i, ..., α p + i ; β + i, β + i, ..., β p + i ; x , (4 . − δ ) i p − F q − α , α , ..., α p − ; β , β , ..., β q − ; x = x i ( − i ( α ) i ( α ) i ... ( α p − ) i ( β ) i ( β ) i ... ( β p − ) i p − F q − α + i, α + i, ..., α p − + i ; β + i, β + i, ..., β p − + i ; x . (4 . p F q α , α , ..., α p ; β , β , ..., β q ; x − h = ∞ X i =0 h i i ! x − i ( − δ ) i p F q α , α , ..., α p ; β , β , ..., β q ; x = ∞ X i =0 ( − i ( α ) i ( α ) i ... ( α p ) i ( β ) i ( β ) i ... ( β q ) i i ! h ip F q α + i, α + i, ..., α p + i ; β + i, β + i, ..., β q + i ; x . Replacing the arguments x, h by (i) x + y , xy and (ii) x + y − xy , − xy , we get (4.5), (4.6). Validity ofexpansion (4.1) - (4.4) can be proved similarly. In the specific case from formulas of expansion (4.1) -(4.4) the following outcomes follow p F q α , α , ..., α p ; β , β , ..., β q ; x = ∞ X j =0 ( α ) j ( α ) j ... ( α p − ) j ( β ) j ( β ) j ... ( β q − ) j j ! x j p F q α + j, α + j, ..., α p − + j, β q − α p ; β + j, β + j, ..., β q + j, β q ; − x , (4 . p F q α , α , ..., α p ; β , β , ..., β q ; x = F p − , , q − , , α , α , ..., α p − ; β , β , ..., β q − ; β q − α p ; β q ; − ; − ; − x, x , (4 . F α , α , α β , β x = (1 − x ) − α F , , , , α ; β ; α , β − α ; β ; β − α ; − ; xx − , xx − , (4 . F p ; q ; kl ; i ; j ( a p ) ;( α l ) ; ( b q ) ;( β m ) ; ( c k ) ;( γ n ) ; x, y = ∞ X r,s =0 p Q j =1 ( a j ) r + s q Q j =1 ( b j ) r k Q j =1 ( c j ) sl Q j =1 ( α j ) r + s m Q j =1 ( β j ) r n Q j =1 ( γ j ) s r ! s ! x r y s . (4 . F a, b, − n ; c, a + b − c − n ; 1 = ( c − a ) n ( c − b ) n ( c ) n ( c − a − b ) n , n ∈ N = { , , ... } , (4 . ∞ X j =0 ( a ) j ( b ) j ( c ) j j ! F a + j, b + j, a + b − c ; c + j, a + b − c − n ; − = ( c − a ) n ( c − b ) n ( c ) n ( c − a − b ) n , n ∈ N, (4 . F α , α , α ; β , β ; x = 1Γ ( α ) ∞ Z e − ξ ξ α − F α , α ; β , β ; xξ dξ, Re α > , (4 . F α , α , α ; β , β ; x = 1Γ ( α ) Γ ( α ) ∞ Z ∞ Z e − ξ e − η ξ α − η α − F α ; β , β ; xξη dξdη, Re α > , Re α > , (4 . F α , α , α ; β , β ; x = 1Γ ( α ) Γ ( α ) Γ ( α ) ∞ Z ∞ Z ∞ Z e − ξ − η − ζ ξ α − η α − ζ α − F − ; β , β ; xξηζ dξdηdζ, Re α i > , i = 1 , , . (4 . As it was marked in the paper [4] to find the formula of an analytic continuation it is necessary to reducethe order of hypergeometric function. For this purpose we use expansion (4.1) at values p = 3 , q = 2.Then the generalized hypergeometric function (1.1) looks like, which refers to as Clausen function ([1],p. 141) F α , α , α ; β , β ; z = ∞ X m =0 ( α ) m ( α ) m ( α ) m ( β ) m ( β ) m m ! z m . (5 . Theorem . If for parameters of Clausen function (5.1) it is satisfied a condition β , β , α − α , β − α − α = 0 , ± , ± , ..., then the following formulas of an analytic continuation are fair F α , α , α ; β , β ; x = B ( − x ) − α F (cid:18) α ; β − α , − β + α ; β , − α + α ; 1 , x (cid:19) + B ( − x ) − α F (cid:18) α ; β − α , − β + α ; β , − α + α ; 1 , x (cid:19) , (5 . F α , α , α ; β , β ; x = B (1 − x ) − α F (cid:18) α ; β − α , β − α ; β , − α − α ; xx − , − x (cid:19) + B (1 − x ) − α F (cid:18) α ; β − α , β − α ; β , − α − α ; xx − , − x (cid:19) , (5 . F α , α , α ; β , β ; x = B F ( α ; α , β − α ; β , − α + α ; x, B F ( α ; α , β − α ; β , − α + α ; x, , (5 . F α , α , α ; β , β ; x = A F , , , , α , α ;1 + α + α − β ; β − α ; β ; − ; − ; x, − x + A (1 − x ) β − α − α H (cid:18) α + α − β ; β − α , β − α , β − α ; β ; xx − , x − (cid:19) , (5 . H ( α ; β, γ, δ ; ε ; x, y ) = ∞ X m,n =0 ( α ) m − n ( β ) m ( γ ) n ( δ ) n ( ε ) m m ! n ! x m y n ,A = Γ ( β ) Γ ( β − α − α )Γ ( β − α ) Γ ( β − α ) , A = Γ ( β ) Γ ( α + α − β )Γ ( α ) Γ ( α ) , (5 . = Γ ( β ) Γ ( α − α )Γ ( α ) Γ ( β − α ) , B = Γ ( β ) Γ ( α − α )Γ ( α ) Γ ( β − α ) . (5 . Proof . Using the formula of an analytic continuation of Gauss function [2] F ( a, b ; c ; x ) = Γ ( c ) Γ ( b − a )Γ ( b ) Γ ( c − a ) ( − x ) − a F (cid:18) a, − c + a ; 1 − b + a ; 1 x (cid:19) + Γ ( c ) Γ ( a − b )Γ ( a ) Γ ( c − b ) ( − x ) − b F (cid:18) b, − c + b ; 1 − a + b ; 1 x (cid:19) , (5 . F α , α , α ; β , β ; x = ( − x ) − α Γ ( β ) Γ ( α − α )Γ ( α ) Γ ( β − α ) ∞ X i =0 ( α ) i ( α ) i ( β − α ) i ( α ) i ( β ) i i ! F (cid:18) α + i, − β + α ; 1 − α + α ; 1 x (cid:19) + ( − x ) − α Γ ( β ) Γ ( α − α )Γ ( α ) Γ ( β − α ) ∞ X i =0 ( α ) i ( α ) i ( β − α ) i ( α ) i ( β ) i i ! F (cid:18) α + i, − β + α ; 1 − α + α ; 1 x (cid:19) . (5 . 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