Applications of an operator H(α,β) to the Lauricella multivariable hypergeometric functions
aa r X i v : . [ m a t h - ph ] O c t Applications of an operator H ( α, β ) to the Lauricellamultivariable hypergeometric functions A. Hasanov
Institute of Mathematics and Information Technologies, Uzbek Academy of Sciences,29, F. Hodjaev street, Tashkent 100125, UzbekistanE-mail: [email protected]
Abstract
By making use of some techniques based upon certain inverse new pairs of symbolic operators,the author investigate several decomposition formulas associated with Lauricella’s hypergeometricfunctions F ( r ) A , F ( r ) B , F ( r ) C and F ( r ) D in r variables. In the three-variable case some of these operationalrepresentations are constructed and applied in order to derive the corresponding decomposition for-mulas when r = 3 . With the help of these new inverse pairs of symbolic operators, a total 20decomposition formulas and integral representations are found. MSC: primary 33C65.Key Words and Phrases.
Decomposition formulas; Lauricella hypergeometric functions; Multiplehypergeometric functions; Generalized hypergeometric functions; Inverse pairs of symbolic operators;Integral representations.
A great interest in the theory of hypergeometric functions (that is, hypergeometric functions of severalvariables) is motivated essentially by the fact that the solutions of many applied problems involving(for example) partial differential equations are obtainable with the help of such hypergeometric function(see, for details, [21, p. 47]; see also the recent works [7-9, 16, 17] and the references cited therein).For instance, the energy absorbed by some nonferromagnetic conductor sphere included in an internalmagnetic field can be calculated with the help of such functions [12, 15]. Hypergeometric functions ofseveral variables are used in physical and quantum chemical applications as well [13, 19 and 20]. Espe-cially, many problems in gas dynamics lead to solutions of degenerate second-order partial differentialequations, which are then solvable in terms of multiple hypergeometric functions. Among examples, wecan cite the problem of adiabatic flat-parallel gas flow without whirlwind, the flow problem of supersoniccurrent from vessel with flat walls, and a number of other problems connected with gas flow [2, 6].We note that Riemann’s functions and fundamental solutions of the degenerate second-order partial dif-ferential equations are expressible by means of hypergeometric functions of several variables [7-9]. Ininvestigation of the boundary value problems for these partial differential equations, we need decompo-sitions for hypergeometric functions of several variables in terms of simpler hypergeometric functions ofthe Gauss and Lauricella types. 1ultiple hypergeometric functions (that is, hypergeometric functions in several variables) occur natu-rally in a wide variety of problems. In particular, the Lauricella functions F ( r ) A , F ( r ) B , F ( r ) C and F ( r ) D in r ( r ∈ \ { } ; := { , , ... } ) variables, defined by ( [1, p. 114]) F ( r ) A ( α ; β , ..., β r ; γ , ..., γ r ; x , ..., x r ) = ∞ P m ,...,m r =0 ( α ) m + ... + m r ( β ) m · · · ( β r ) m r ( γ ) m · · · ( γ r ) m r m ! · · · m r ! x m · · · x m r r , ( | x | + · · · + | x r | < , (1 . F ( r ) B ( α , ..., α r ; β , ..., β r ; γ ; x , ..., x r ) = ∞ P m ,...,m r =0 ( α ) m · · · ( α r ) m r ( β ) m · · · ( β r ) m r ( γ ) m + ... + m r m ! · · · m r ! x m · · · x m r r , (max [ | x | , ..., | x r | ] < , (1 . F ( r ) C ( α, β ; γ , ..., γ r ; x , ..., x r ) = ∞ P m ,...,m r =0 ( α ) m + ... + m r ( β ) m + ... + m r ( γ ) m · · · ( γ r ) m r m ! · · · m r ! x m · · · x m r r , (cid:0)(cid:12)(cid:12) √ x (cid:12)(cid:12) + · · · + (cid:12)(cid:12) √ x r (cid:12)(cid:12) < (cid:1) , (1 . F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r ) = ∞ X m ,...,m r =0 ( α ) m + ... + m r ( β ) m · · · ( β r ) m r ( γ ) m + ... + m r m ! · · · m r ! x m · · · x m r r , (max [ | x | , ..., | x r | ] < . (1 . α ) m = Γ ( α + m ) / Γ ( α ) denotes the Pochhammer symbol (or the shiftedfactorial) for all admissible (real or complex) values of α and m .For various multivariable hypergeometric functions including the Lauricella multivariable functions F ( r ) A , F ( r ) B , F ( r ) C and F ( r ) D , defined by (1.1)-(1.4), Hasanov and Srivastava [10, 11] presented a numberof decompositions formulas in terms of such simpler hypergeometric functions as the Gauss and Appellfunctions. The main object of this sequel to the works of Hasanov and Srivastava [10, 11] is to show howsome rather elementary techniques based upon certain inverse pairs of symbolic operators would lead useasily to several decomposition formulas associated with Lauricella’s hypergeometric function F ( r ) A , F ( r ) B , F ( r ) C and F ( r ) D in r variables and with other multiple hypergeometric functions.Over six decades ago, Burchnall and Chaundy [3, 4] and Chaundy [5] systematically presented anumber of expansion and decomposition formulas for some double hypergeometric functions in series ofsimpler hypergeometric functions. Their method is based upon the following inverse pairs of symbolicoperators: ∇ xy ( h ) := Γ ( h ) Γ ( δ + δ + h )Γ ( δ + h ) Γ ( δ + h ) = ∞ X k =0 ( − δ ) k ( − δ ) k ( h ) k k ! , (1 . xy ( h ) := Γ ( δ + h ) Γ ( δ + h )Γ ( h ) Γ ( δ + δ + h ) = ∞ X k =0 ( − δ ) k ( − δ ) k (1 − h − δ − δ ) k k != ∞ X k =0 ( − k ( h ) k ( − δ ) k ( − δ ) k ( h + k − k ( h + δ ) k ( h + δ ) k k ! , (1 . ∇ xy ( h ) ∆ xy ( g ) := Γ ( h ) Γ ( δ + δ + h )Γ ( δ + h ) Γ ( δ + h ) Γ ( δ + g ) Γ ( δ + g )Γ ( g ) Γ ( δ + δ + g )= ∞ X k =0 ( g − h ) k ( g ) k ( − δ ) k ( − δ ) k ( g + k − k ( g + δ ) k ( g + δ ) k k != ∞ X k =0 ( h − g ) k ( − δ ) k ( − δ ) k ( h ) k (1 − g − δ − δ ) k k ! (cid:18) δ := x ∂∂x ; δ := y ∂∂y (cid:19) . (1 . 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 ! (1 . 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 ! , (1 . (cid:18) δ j := x j ∂∂x j , j = 1 , ..., l ; l ∈ N := { , , , ... } (cid:19) where we have applied known multiple hypergeometric summation formulas as follows [1, p. 117]: F ( r ) D ( α ; β , ..., β r ; γ ; 1 , ...,
1) = Γ ( γ ) Γ ( γ − α − β − · · · − β r )Γ ( γ − α ) Γ ( γ − β − · · · − β r ) (cid:0) Re ( γ − α − β − · · · − β r ) > , γ / ∈ N − := { , − , − , − , ... } (cid:1) for the Lauricella function F ( r ) D in r variables, defined by (1.4). First of all, it is not difficult to derive the following applications of the (multivariable) symbolic operatorsdefined by (1.8) and (1.9): F ( r ) A ( α ; β , ..., β r ; γ , ..., γ r ; x , ..., x r ) = H x ,...,x r ( α, ε ) F ( r ) A ( ε ; β , ..., β r , γ , ..., γ r ; x , ..., x r ) , (2 . F ( r ) A ( α ; β , ..., β r ; γ , ..., γ r ; x , ..., x r ) = ¯ H x ,...,x r ( ε, α ) F ( r ) A ( ε ; β , ..., β r , γ , ..., γ r ; x , ..., x r ) , (2 . F ( r ) B ( α , ..., α r ; β , ..., β r , γ ; x , ..., x r ) = ¯ H x ,...,x r ( γ, ε ) F ( r ) B ( α , ..., α r ; β , ..., β r , ε ; x , ..., x r ) , (2 . F ( r ) C ( α, β ; γ , ..., γ r ; x , ..., x r ) = H x ,...,x r ( α, ε ) F ( r ) C ( ε, β ; γ , ..., γ r ; x , ..., x r ) , (2 . F ( r ) C ( α, β ; γ , ..., γ r ; x , ..., x r ) = ¯ H x ,...,x r ( ε, α ) F ( r ) C ( ε, β ; γ , ..., γ r ; x , ..., x r ) , (2 . F ( r ) C ( α, β ; γ , ..., γ r ; x , ..., x r ) = H x ,...,x r ( α, ε ) H x ,...,x r ( β, ε ) F ( r ) C ( ε , ε ; γ , ..., γ r ; x , ..., x r ) , (2 . F ( r ) C ( α, β ; γ , ..., γ r ; x , ..., x r ) = ¯ H x ,...,x r ( ε , α ) ¯ H x ,...,x r ( ε , β ) F ( r ) C ( ε , ε ; γ , ..., γ r ; x , ..., x r ) , (2 . F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r ) = H x ,...,x r ( α, ε ) F ( r ) D ( ε ; β , ..., β r ; γ ; x , ..., x r ) , (2 . F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r ) = ¯ H x ,...,x r ( ε, α ) F ( r ) D ( ε ; β , ..., β r ; γ ; x , ..., x r ) , (2 . F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r ) = H x ,...,x r ( ε, γ ) F ( r ) D ( α ; β , ..., β r ; ε ; x , ..., x r ) , (2 . F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r ) = H x ,...,x r ( α, γ ) (1 − x ) − β · · · (1 − x r ) − β r , (2 . − x ) − β · · · (1 − x r ) − β r = ¯ H x ,...,x r ( α, γ ) F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r ) . (2 . H x ,...,x r ( α, ε ) H x ,...,x r ( β, ε )= ∞ X k , ··· ,k r ,l , ··· ,l r =0 ( ε − α ) k + ··· + k r ( ε − β ) l + ··· + l r ( β ) k + ··· + k r ( − δ ) k + l · · · ( − δ r ) k r + l r ( ε ) k + ··· + k r ( ε ) k + ··· + k r + l + ··· + l r k ! · · · k r ! l ! · · · l r ! (2 . H x ,...,x r ( ε , α ) ¯ H x ,...,x r ( ε , β )= ∞ X k ,...k r ,l ,...l r =0 ( − k + ··· + k r ( α − ε ) k + ··· + k r ( β − ε ) k + ··· + k r + l + ··· + l r ( β ) k + ... + k r ( β − ε ) k + ··· + k r k ! · · · k r ! l ! · · · l r ! × ( − δ ) k + l · · · ( − δ r ) k r + l r (1 − ε − δ − · · · − δ r ) k + ··· + k r (1 − ε − δ − · · · − δ r ) k + ··· + k r + l + ··· + l r . (2 . F ( r ) A , F ( r ) B , F ( r ) C and F ( r ) D , it is not difficult to give alternative proofs of the operator identities (2.1) to (2.12) aboveby using the Mellin and the inverse Mellin transformations (see, for example, [14]). The details involvedin these alternative derivations of the operator identities (2.1) to (2.12) are being omitted here.By virtue of the derivative formulas for the Lauricella functions, and also of some standard propertiesof hypergeometric functions, we find each of the following decompositions formulas for the Lauricellafunctions F ( r ) A , F ( r ) B , F ( r ) C and F ( r ) D : F ( r ) A ( α ; β , ..., β r ; γ , ..., γ r ; x , ..., x r ) = ∞ X k , ··· ,k r =0 ( − k + ··· + k r ( ε − α ) k + ··· + k r ( β ) k · · · ( β r ) k r ( γ ) k · · · ( γ r ) k r k ! · · · k r ! × x k x k · · · x k r r F ( r ) A ( ε + k + · · · + k r ; β + k , ..., β r + k r ; γ + k , ..., γ r + k r ; x , ..., x r ) , (2 . F ( r ) A ( α ; β , ..., β r ; γ , ..., γ r ; x , ..., x r ) = ∞ X k ,...k r =0 ( α − ε ) k + ··· + k r ( β ) k ( β ) k · · · ( β r ) k r ( γ ) k ( γ ) k · · · ( γ r ) k r k ! · · · k r ! × x k · · · x k r r F ( r ) A ( ε ; β + k , ..., β r + k r ; γ + k , ..., γ r + k r ; x , ..., x r ) , (2 . F ( r ) B ( α , ..., α r ; β , ..., β r ; γ ; x , ..., x r )= ∞ X k , ··· ,k r =0 ( − k + ··· + k r ( γ − ε ) k + ··· + k r ( α ) k · · · ( α r ) k r ( β ) k · · · ( β r ) k r ( γ ) k + ··· + k r ( ε ) k + ··· + k r k ! · · · k r ! × x k · · · x k r r F ( r ) B ( α + k , ..., α r + k r ; β + k , ..., β r + k r ; ε + k + · · · + k r ; x , ..., x r ) , (2 . F ( r ) C ( α, β ; γ , ..., γ r ; x , ..., x r ) = ∞ X k , ··· ,k r =0 ( − k + ··· + k r ( ε − α ) k + ··· + k r ( β ) k + ··· + k r ( γ ) k · · · ( γ r ) k r k ! · · · k r ! x k · · · x k r r × F ( r ) C ( ε + k + · · · + k r , β + k + · · · + k r ; γ + k , ..., γ r + k r ; x , ..., x r ) , (2 . F ( r ) C ( α, β ; γ , ..., γ r ; x , ..., x r ) = ∞ X k ,...k r =0 ( α − ε ) k + ··· + k r ( β ) k + ··· + k r ( γ ) k · · · ( γ r ) k r k ! · · · k r ! x k · · · x k r r × F ( r ) C ( ε, β + k + · · · + k r ; γ + k , ..., γ r + k r ; x , ..., x r ) , (2 . F ( r ) C ( α, β ; γ , ..., γ r ; x , ..., x r ) = ∞ X k , ··· ,k r ,l , ··· ,l r =0 ( − k + ··· + k r + l + ··· + l r ( ε − α ) k + ··· + k r ( ε ) k + ··· + k r ( γ ) k + l · · · ( γ r ) k r + l r ( ε − β ) l + ··· + l r ( β ) k + ··· + k r ( ε ) k + ··· + k r + l + ··· + l r k ! · · · k r ! l ! · · · l r ! x k + l · · · x k r + l r r × F ( r ) C ( ε + k + · · · + k r + l + · · · + l r , ε + k + · · · + k r + l + · · · + l r ; γ + k + l , ..., γ r + k r + l r ; x , ..., x r ) , (2 . ( r ) C ( α, β ; γ , ..., γ r ; x , ..., x r )= ∞ X k ,...k r ,l ,...l r =0 ( − l + ··· + l r ( α − ε ) k + ··· + k r ( β − ε ) k + ··· + k r + l + ··· + l r ( β ) k + ... + k r ( β − ε ) k + ··· + k r ( γ ) k + l · · · ( γ r ) k r + l r k ! · · · k r ! l ! · · · l r ! x k + l · · · x k r + l r r × F ( r ) C ( ε + l + · · · + l r , ε + l + · · · + l r ; γ + k + l , ..., γ r + k r + l r ; x , ..., x r ) , (2 . F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r ) = ∞ X k , ··· ,k r =0 ( − k + ··· + k r ( ε − α ) k + ··· + k r ( β ) k · · · ( β ) k ( γ ) k + ··· + k r k ! · · · k r ! x k · · · x k r r × F ( r ) D ( ε + k + · · · + k r ; β + k , ..., β r + k r ; γ + k + · · · + k r ; x , ..., x r ) , (2 . F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r ) = ∞ X k ,...k r =0 ( α − ε ) k + ··· + k r ( β ) k · · · ( β r ) k r ( γ ) k + ··· + k r k ! · · · k r ! x k · · · x k r r × F ( r ) D ( ε ; β + k , ..., β r + k r ; γ + k + · · · + k r ; x , ..., x r ) , (2 . F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r )= ∞ X k , ··· ,k r =0 ( − k + ··· + k r ( γ − ε ) k + ··· + k r ( α ) k + ··· + k r ( β ) k · · · ( β r ) k r ( γ ) k + ··· + k r ( ε ) k + ··· + k r k ! · · · k r ! x k · · · x k r r × F ( r ) D ( α + k + · · · + k r ; β + k , ..., β r + k r ; ε + k + · · · + k r ; x , ..., x r ) , (2 . F ( r ) D ( α ; β , ..., β r ; γ ; x , ..., x r )= (1 − x ) − β · · · (1 − x r ) − β r F ( r ) D (cid:18) γ − α ; β , ..., β r ; γ ; x x − , ..., x r x r − (cid:19) , (2 . − x ) − β · · · (1 − x r ) − β r = ∞ X k ,...k r =0 ( γ − α ) k + ··· + k r ( β ) k · · · ( β r ) k r ( γ ) k + ··· + k r k ! · · · k r ! x k · · · x k r r × F ( r ) D ( α ; β + k , ..., β r + k r ; γ + k + · · · + k r ; x , ..., x r ) , (2 . δ + α ) n { f ( ξ ) } = ξ − α d n dξ n (cid:8) ξ α + n − f ( ξ ) (cid:9) (2 . (cid:18) δ := ξ ddξ ; α ∈ ; n ∈ := ∪ { } ; := { , , , ... } (cid:19) and ( − δ ) n { f ( ξ ) } = ( − n ξ n d n dξ n { f ( ξ ) } , (cid:18) δ := ξ ddξ ; n ∈ (cid:19) (2 . f ( ξ ). Many other analogous decomposition formulas can similarly be derivedfor the Lauricella functions F ( r ) A , F ( r ) B , F ( r ) C and F ( r ) D in r variables, but with various different parametricconstraints. In this section, we consider the Lauricella functions F (3) A , F (3) B , F (3) C and F (3) D in three variables. Wereiterate the aforementioned fact that numerous other analogous decompositions formulas can similarly5e derived for the Lauricella triple hypergeometric functions F (3) A , F (3) B , F (3) C and F (3) D with differentparametric constrains. For example, each of the following operational representations would lead us adecomposition formula for a special F (3) A , F (3) B , F (3) C and F (3) D in three variables. F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = H x ( β , ε ) F A ( α ; ε , β , β ; γ , γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = ¯ H x ( ε , β ) F A ( α ; ε , β , β ; γ , γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = H x ( β , ε ) H y ( β , ε ) F A ( α ; ε , ε , β ; γ , γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = ¯ H x ( ε , β ) ¯ H y ( ε , β ) F A ( α ; ε , ε , β ; γ , γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = H x ( β , ε ) H y ( β , ε ) H z ( β , ε ) F A ( α ; ε , ε , ε ; γ , γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = ¯ H x ( ε , β ) ¯ H y ( ε , β ) ¯ H z ( ε , β ) F A ( α ; ε , ε , ε ; γ , γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = H x ( β , γ ) (1 − x ) − α F (cid:18) α ; β , β ; γ , γ ; y − x , z − x (cid:19) , (3 . − x ) − α F (cid:18) α ; β , β ; γ , γ ; y − x , z − x (cid:19) = ¯ H x ( β , γ ) F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = H x ( β , γ ) H y ( β , γ ) (1 − x − y ) − α F (cid:18) α, β ; γ ; z − x − y (cid:19) , (3 . − x − y ) − α F (cid:18) α, β ; γ ; z − x − y (cid:19) = ¯ H x ( β , γ ) ¯ H y ( β , γ ) F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = H x ( β , γ ) H y ( β , γ ) H z ( β , γ ) (1 − x − y − z ) − α , (3 . − x − y − z ) − α = ¯ H x ( β , γ ) ¯ H y ( β , γ ) ¯ H z ( β , γ ) F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z ) = H x ( α , ε ) F B ( ε , α , α ; β , β , β ; γ ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z ) = ¯ H x ( ε , α ) F B ( ε , α , α ; β , β , β ; γ ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z ) = H x ( α , ε ) H y ( α , ε ) F B ( ε , ε , α ; β , β , β ; γ ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z ) = ¯ H x ( ε , α ) ¯ H y ( ε , α ) F B ( ε , ε , α ; β , β , β ; γ ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z )= H x ( α , ε ) H y ( α , ε ) H z ( α , ε ) F B ( ε , ε , ε ; β , β , β ; γ ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z )= ¯ H x ( ε , α ) ¯ H y ( ε , α ) ¯ H z ( ε , α ) F B ( ε , ε , ε ; β , β , β ; γ ; x, y, z ) , (3 . F C ( α, β ; γ , γ , γ ; x, y, z ) = H x ( ε , γ ) F C ( α, β ; ε , γ , γ ; x, y, z ) , (3 . F C ( α, β ; γ , γ , γ ; x, y, z ) = H x ( ε , γ ) H y ( ε , γ ) F C ( α, β ; ε , ε , γ ; x, y, z ) , (3 . F C ( α, β ; γ , γ , γ ; x, y, z ) = H x ( ε , γ ) H y ( ε , γ ) H z ( ε , γ ) F C ( α, β ; ε , ε , ε ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z ) = H x ( β , ε ) F D ( α ; ε , β , β ; γ ; x, y, z ) , (3 . D ( α ; β , β , β ; γ ; x, y, z ) = ¯ H x ( ε , β ) F D ( α ; ε , β , β ; γ ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z ) = H x ( β , ε ) H y ( β , ε ) F D ( α ; ε , ε , β ; γ ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z ) = ¯ H x ( ε , β ) ¯ H y ( ε , β ) F D ( α ; ε , ε , β ; γ ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z ) = H x ( β , ε ) H y ( β , ε ) H z ( β , ε ) F D ( α ; ε , ε , ε ; γ ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z ) = ¯ H x ( ε , β ) ¯ H y ( ε , β ) ¯ H z ( ε , β ) F D ( α ; ε , ε , ε ; γ ; x, y, z ) , (3 . F A := F (3) A , F B := F (3) B , F C := F (3) C , F C := F (3) C , F := F (2) A . (3 . F (3) A , F (3) B , F (3) C and F (3) D : F A ( α ; β , β , β ; γ , γ , γ ; x, y, z )= ∞ X i =0 ( − i ( ε − β ) i ( α ) i ( γ ) i i ! x i F A ( α + i ; ε + i, β , β ; γ + i, γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = ∞ X i =0 ( β − ε ) i ( α ) i ( γ ) i i ! x i F A ( α + i ; ε , β , β ; γ + i, γ , γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = ∞ X i,j =0 ( − i + j ( α ) i + j ( ε − β ) i ( ε − β ) j ( γ ) i ( γ ) j i ! j ! x i y j × F A ( α + i + j ; ε + i, ε + j, β ; γ + i, γ + j, γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = ∞ X i,j =0 ( α ) i + j ( β − ε ) i ( β − ε ) j ( γ ) i ( γ ) j i ! j ! x i y j × F A ( α + i + j ; ε , ε , β ; γ + i, γ + j, γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z )= ∞ X i,j,k =0 ( − i + j + k ( α ) i + j + k ( ε − β ) i ( ε − β ) j ( ε − β ) k ( γ ) i ( γ ) j ( γ ) k i ! j ! k ! x i y j z k × F A ( α + i + j + k ; ε + i, ε + j, ε + k ; γ + i, γ + j, γ + k ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = ∞ X i,j,k =0 ( α ) i + j + k ( β − ε ) i ( β − ε ) j ( β − ε ) k ( γ ) i ( γ ) j ( γ ) k i ! j ! k ! x i y j z k × F A ( α + i + j + k ; ε , ε , ε ; γ + i, γ + j, γ + k ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z )= (1 − x ) − α ∞ X i =0 ( α ) i ( γ − β ) i ( γ ) i i ! (cid:18) xx − (cid:19) i F (cid:18) α + i ; β , β ; γ , γ ; y − x , z − x (cid:19) , (3 . − x ) − α F (cid:18) α ; β , β ; γ , γ ; y − x , z − x (cid:19) = ∞ X i =0 ( α ) i ( γ − β ) i ( γ ) i i ! x i F A ( α + i ; β , β , β ; γ + i, γ , γ ; x, y, z ) , (3 . A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = (1 − x − y ) − α ∞ X i,j =0 ( − i + j ( α ) i + j ( γ − β ) i ( γ − β ) j ( γ ) i ( γ ) j i ! j ! × (cid:18) x − x − y (cid:19) i (cid:18) y − x − y (cid:19) j F (cid:18) α + i + j, β ; γ ; z − x − y (cid:19) , (3 . − x − y ) − α F (cid:18) α, β ; γ ; z − x − y (cid:19) = ∞ X i,j =0 ( α ) i + j ( γ − β ) i ( γ − β ) j ( γ ) i ( γ ) j i ! j ! x i y j F A ( α + i + j ; β , β , β ; γ + i, γ + j, γ ; x, y, z ) , (3 . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = (1 − x − y − z ) − α × F A (cid:18) α ; γ − β , γ − β , γ − β ; γ , γ , γ ; xx + y + z − , yx + y + z − , zx + y + z − (cid:19) , (3 . − x − y − z ) − α = ∞ X i,j,k =0 ( α ) i + j + k ( γ − β ) i ( γ − β ) j ( γ − β ) k ( γ ) i ( γ ) j ( γ ) k i ! j ! k ! x i y j z k × F A ( α + i + j + k ; β , β , β ; γ + i, γ + j, γ + k ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z )= ∞ X i =0 ( − i ( ε − α ) i ( β ) i ( γ ) i i ! x i F B ( ε + i, α , α ; β + i, β , β ; γ + i ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z )= ∞ X i =0 ( α − ε ) i ( β ) i ( γ ) i i ! x i F B ( ε , α , α ; β + i, β , β ; γ + i ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z ) = ∞ X i,j =0 ( − i + j ( ε − α ) i ( ε − α ) j ( β ) i ( β ) j ( γ ) i + j i ! j ! x i y j × F B ( ε + i, ε + j, α ; β + i, β + j, β ; γ + i + j ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z )= ∞ X i,j =0 ( α − ε ) i ( α − ε ) i ( β ) i ( β ) j ( γ ) i + j i ! j ! x i y j F B ( ε , ε , α ; β + i, β + j, β ; γ + i + j ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z )= ∞ X i,j,k =0 ( − i + j + k ( ε − α ) i ( ε − α ) j ( ε − α ) k ( β ) i ( β ) j ( β ) k ( γ ) i + j + k i ! j ! k ! x i y j z k × F B ( ε + i, ε + j, ε + k ; β + i, β + j, β + k ; γ + i + j + k ; x, y, z ) , (3 . F B ( α , α , α ; β , β , β ; γ ; x, y, z )= ∞ X i,j,k =0 ( α − ε ) i ( α − ε ) i ( α − ε ) k ( β ) i ( β ) j ( β ) k ( γ ) i + j + k i ! j ! k ! x i y j z k × F B ( ε , ε , ε ; β + i, β + j, β + k ; γ + i + j + k ; x, y, z ) , (3 . F C ( α, β ; γ , γ , γ ; x, y, z ) = ∞ X i =0 ( − i ( α ) i ( β ) i ( γ − ε ) i ( γ ) i ( ε ) i i ! x i F C ( α + i, β + i ; ε + i, γ , γ ; x, y, z ) , (3 . F C ( α, β ; γ , γ , γ ; x, y, z ) = ∞ X i,j =0 ( − i + j ( α ) i + j ( β ) i + j ( γ − ε ) i ( γ − ε ) j ( γ ) i ( γ ) j ( ε ) i ( ε ) j i ! j ! x i y j × F C ( α + i + j, β + i + j ; ε + i, ε + j, γ ; x, y, z ) , (3 . C ( α, β ; γ , γ , γ ; x, y, z )= ∞ X i,j,k =0 ( − i + j + k ( α ) i + j + k ( β ) i + j + k ( γ − ε ) i ( γ − ε ) j ( γ − ε ) k ( γ ) i ( γ ) j ( γ ) k ( ε ) i ( ε ) j ( ε ) k i ! j ! k ! x i y j z k × F C ( α + i + j + k, β + i + j + k ; ε + i, ε + j, ε + k ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z ) = ∞ X i =0 ( − i ( α ) i ( ε − β ) i ( γ ) i i ! x i F D ( α + i ; ε + i, β , β ; γ + i ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z ) = ∞ X i =0 ( α ) i ( β − ε ) i ( γ ) i i ! x i F D ( α + i ; ε , β , β ; γ + i ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z )= ∞ X i,j =0 ( − i + j ( α ) i + j ( ε − β ) i ( ε − β ) j ( γ ) i + j i ! j ! x i y j F D ( α + i + j ; ε + i, ε + j, β ; γ + i + j ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z )= ∞ X i,j =0 ( α ) i + j ( β − ε ) i ( β − ε ) j ( γ ) i + j i ! j ! x i y j F D ( α + i + j ; ε , ε , β ; γ + i + j ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z ) = ∞ X i,j,k =0 ( − i + j + k ( α ) i + j + k ( ε − β ) i ( ε − β ) j ( ε − β ) k ( γ ) i + j + k i ! j ! k ! x i y j z k × F D ( α + i + j + k ; ε + i, ε + j, ε + k ; γ + i + j + k ; x, y, z ) , (3 . F D ( α ; β , β , β ; γ ; x, y, z ) = ∞ X i,j,k =0 ( α ) i + j + k ( β − ε ) i ( β − ε ) j ( β − ε ) k ( γ ) i + j + k i ! j ! k ! x i y j z k × F D ( α + i + j + k ; ε , ε , ε ; γ + i + j + k ; x, y, z ) . (3 . The various decomposition formulas for the Lauricella functions F A , F B , F C and F D in three variables(which are presented here and in other places in the previously cited literature) can be proven fairly simplyby suitably applying superposition of the inverse pairs of symbolic operators introduced in Section 1. Asan example, we shall briefly indicate the proof of the decomposition formula (3.54). For the three-variableLauricella function F D , it is not difficult to show from (3.26) that F D ( α ; β , β , β ; γ ; x, y, z )= ∞ X i =0 ( ε − β ) i ( − δ ) i ( ε ) i i ! ∞ X j =0 ( ε − β ) j ( − δ ) j ( ε ) j j ! ∞ X k =0 ( ε − β ) k ( − δ ) k ( ε ) k k ! F D ( α ; ε , ε , ε ; γ ; x, y, z ) , (4 . (cid:18) δ := x ∂∂x ; δ := y ∂∂y ; δ := z ∂∂z (cid:19) . Furthermore, by a straightforward computation, we have( − δ ) i F D ( α ; ε , ε , ε ; γ ; x, y, z ) = ( − i x i ( α ) i ( ε ) i ( γ ) i F D ( α + i ; ε + i, ε , ε ; γ + i ; x, y, z ) , (4 . − δ ) j ( − δ ) i F D ( α ; ε , ε , ε ; γ ; x, y, z )= ( − i + j x i y j ( α ) i + j ( ε ) i ( ε ) j ( γ ) i + j F D ( α + i + j ; ε + i, ε + j, ε ; γ + i + j ; x, y, z ) (4 . − δ ) k ( − δ ) j ( − δ ) i F D ( α ; ε , ε , ε ; γ ; x, y, z ) = ( − i + j + k x i y j z k × ( α ) i + j + k ( ε ) i ( ε ) j ( ε ) k ( γ ) i + j + k F D ( α + i + j + k ; ε + i, ε + j, ε + k ; γ + i + j + k ; x, y, z ) . (4 . Here in this section, we observe that several integral representations of the Eulerian type can be de-duced also from the corresponding decomposition formulas of Section 3. For example, using integralrepresentation ([1], p. 115, (5)) F ( r ) A ( α ; β , β , β , γ , γ , γ ; x, y, z ) = Γ ( γ ) Γ ( γ ) Γ ( γ )Γ ( β ) Γ ( β ) Γ ( β ) Γ ( γ − β ) Γ ( γ − β ) Γ ( γ − β ) R ... R t β − t β − t β − (1 − t ) γ − β − (1 − t ) γ − β − (1 − t ) γ − β − (1 − xt − yt − zt ) − α dt dt dt , Re γ l > Re β l > , l = 1 , , . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = Γ ( γ ) Γ ( γ ) Γ ( γ )Γ ( ε ) Γ ( β ) Γ ( β ) Γ ( γ − ε ) Γ ( γ − β ) Γ ( γ − β ) Z ... Z t ε − t β − t β − (1 − t ) γ − ε − (1 − t ) γ − β − (1 − t ) γ − β − × (1 − yt − zt ) − α F (cid:18) α, β ; ε ; xt − yt − zt (cid:19) dt dt dt , Re γ > Re ε > , Re γ l > Re β l > , l = 2 , . F A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = Γ ( γ ) Γ ( γ ) Γ ( γ )Γ ( ε ) Γ ( ε ) Γ ( β ) Γ ( γ − ε ) Γ ( γ − ε ) Γ ( γ − β ) × Z ... Z t ε − t ε − t β − (1 − t ) γ − ε − (1 − t ) γ − ε − (1 − t ) γ − β − (1 − xt − yt − zt ) − α × F (cid:18) α ; β − ε , β − ε ; γ − ε , γ − ε ; x (1 − t )1 − xt − yt − zt , y (1 − t )1 − xt − yt − zt (cid:19) dt dt dt Re γ l > Re ε l > , l = 1 , , Re γ > Re β > , (5 . A ( α ; β , β , β ; γ , γ , γ ; x, y, z ) = Γ ( γ ) Γ ( γ ) Γ ( γ )Γ ( ε ) Γ ( ε ) Γ ( ε ) Γ ( γ − ε ) Γ ( γ − ε ) Γ ( γ − ε ) × Z ... Z t ε − t ε − t ε − (1 − t ) γ − ε − (1 − t ) γ − ε − (1 − t ) γ − ε − (1 − xt − yt − zt ) − α × F A ( α ; β − ε , β − ε , β − ε ; γ − ε , γ − ε , γ − ε ; x (1 − t )1 − xt − yt − zt , y (1 − t )1 − xt − yt − zt , z (1 − t )1 − xt − yt − zt (cid:19) dt dt dt , Re γ l > Re ε l > , l = 1 , , . (5 . Remark.
Introduced operators can be used for other hypergeometric functions also.
References [1] P. Appell and J. Kampe de Feriet,
Fonctions Hypergeometriques et Hyperspheriques; Polynomesd’Hermite , Gauthier - Villars. Paris, 1926.[2] L. Bers,
Mathematical Aspects of Subsonic and Transonic Gas Dynamics , Wiley, New York, 1958.[3] J.L. Burchnall, T.W. Chaundy, Expansions of Appell’s double hypergeometric functions,
Quart. J.Math.
Oxford Ser. 11 (1940), 249-270.[4] J.L. Burchnall, T.W. Chaundy, Expansions of Appell’s double hypergeometric functions. II,
Quart.J. Math.
Oxford Ser. 12 (1941), 112-128.[5] T.W. Chaundy, Expansions of hypergeometric functions,
Quart. J. Math.
Oxford Ser. 13, (1942),159-171.[6] F.I. Frankl,
Selected Works in Gas Dynamics . Nauka, Moscow 1973 (in Russian).[7] A. Hasanov, Fundamental solutions of generalized bi-axially symmetric Helmholtz equation.
ComplexVariables and Elliptic Equations . 52 (8) (2007) 673-683.[8] A. Hasanov, The solution of the Cauchy problem for generalized Euler-Poisson-Darboux equation.
International Journal of Applied Mathematics and Statistics . 8 (7) (2007) 30-44.[9] A. Hasanov, On a mixed problem for the equation sign y | y | m u xx + x n u yy = 0. Izv. Akad. NaukUzSSR , ser. Fiz.-mat. Nauk, 2 (1982) 28-32. (in Russian)[10] A. Hasanov and H. M. Srivastava, Some decomposition formulas associated with the Lauricellafunction F ( r ) A and other multiple hypergeometric functions, App. Math. Lett . 19 (2006) 113-121.[11] A. Hasanov and H.M. Srivastava, Decomposition Formulas Associated with the Lauricella Multi-variable Hypergeometric Functions.
Computers and Mathematics with Applications , 53 (7) (2007)1119-1128. 1112] G. Lohofer, Theory of an electromagnetically deviated metal sphere. 1: Absorbed power.
SIAM J.Appl. Math.
49 (1989) 567-581.[13] A.M. Mathai, R. K. Saxena,
Generalized Hypergeometric Functions with Applications in Statisticsand Physical Sciences . Springer-Verlag, Berlin, Heidelberg and New York. 1973.[14] O.I. Marichev,
Handbook of Integral Transforms of Higher Transcendental Functions: Theory andalgorithmic Tables . Halsted Press (Ellis Horwood Limited, Chichester), Wiley, New York, Chichester,Brisbane and Toronto, 1982.[15] A.W. Niukkanen, Generalised hypergeometric series N F ( x , ..., x N ) arising in physical and quantumchemical applications, J. Phys. A: Math. Gen.
16 (1983) 1813-1825.[16] S.B. Opps, N. Saad, H.M. Srivastava, Some reduction and transformation formulas for the Appellhypergeometric function F , J. Math. Anal. Appl.
302 (2005) 180-195.[17] P.A. Padmanabham, H.M. Srivastava, Summation formulas associated with the Lauricella function F ( r ) A , Appl. Math. Lett.
13 (1) (2000) 65-70.[18] E.G. Poole,
Introduction to the Theory of Linear Differential Equations , Clarendon (Oxford Univer-sity Press), Oxford, 1936.[19] I.N. Sneddon.
Special Functions of Mathematical Physics and Chemistry.
Third ed., Longman, Lon-don and New York. 1980.[20] H.M. Srivastava., B.R.K. Kashyap.
Special Functions in Queuing Theory and Related StochasticProcesses . Academic Prees, New York, London and San Francisco, 1982.[21] H.M. Srivastava, P.W. Karlsson,