APPELL S AND HUMBERT S FUNCTIONS OF MATRIX ARGUMENTS I

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1 AELL S AND HMBER S NCIONS O MARI ARGMENS I Lalit Mohan adhyaya* & H. S. Dhami** Deartment of Mathematics, niversity of Kumaun, Almora Camus, Almora ttaranchal, India 66. AMS Mathematics Subject Classification : rimary: C65, C99. Secondary: 6E, 6H, A5. Key Words : Aell s functions, Humbert s functions, matrix arguments, matrixtransform. ABSRAC We have roved six results for the Aell s functions of matrix arguments-two each for the functions and and one each for the functions and. INRODCION Aell s functions of matrix arguments have earlier been studied by Mathai [, 5, 6] and also by Saxena, Sethi and Guta [7]. In the resent aer we have utilized Mathai s definitions for all the functions studied. All the matrices aearing in this aer are x real symmetric ositive definite matrices and the meanings of all the other symbols used are the same as in the works of Mathai [,].. reliminary Definitions DEINIION.: he Aell s function a,b,b ;c;, of matrix arguments is defined as that function for which the M-transform matrix-transform is the following: * Deartment of Mathematics, Municial ost Graduate College, Mussoorie, Dehradun ttaranchal, India, 879. ** o whom all the corresondence may be addressed.

2 M > > c / / a b a b b c a b b c a, b, b ;c;, dd b. for Re a, b, b, c,, /. > DEINIION.: a, b, b ;c, c ;, M + / + / a, b, b ;c, c ;, dd > > c c a b b. for Re a, b, b, c, c,, > /. DEINIION.: a, a, b, b ;c;, M + / + / a, a, b, b ;c;, dd > > c c a a b a a b b c. for Re a, a, b, b,c,, > /. DEINIION.: a, b;c, c ;, M / / > > a, b;c, c ;, dd c c a b for Re a, b a b c, c, c c,, > b /..

3 DEINIION.5: he Humbert s function Φ a, b; c;, Φ of matrix arguments is defined as that function which has the following M-transform: M / / Φ a, b;c;, dd > > Φ c a b.5 a b c for Re a, b, c,, > /. DEINIION.6: a, b;c,c ;, M / / > > c c a a b c.6 for Re a, b,c, c,, > /. DEINIION.7: a;c,c ;, M / / a;c, c ;, dd > > c c a.7 a c c c a, b;c,c b ;, dd for Re a,c,c,, > /. DEINIION.8: Ξ a, b;c;, Ξ M / / Ξ a, b;c;, dd > > Ξ c a b.8 a b c for Re a, b,c,, > /.

4 HEOREM.:,, ; ;, α β β γ γ β β γ β β I γββ +. Aell s unctions of Matrix Arguments. / I I β+β + / V β + / I / V β I + I V I V + V V α ddv. for < < I, < V < I and for Re β, β, γ β β > /. ROO: We take the M-transform of the right side of eq.. with resect to the variables and and the arameters and resectively to obtain, + / + / I I V I V > > + + V V α dd. Making use of the transformations, I V I d I V + / + d, d I V, V in the above exression and then integrating out the variables tye- Dirichlet integral we get, / V, V V V + / + / and with, d and by using a

5 5 α V I V. α Substituting this exression on the right side of eq.. and then integrating out the variables and V in the resulting exression by using a tye- Beta integral generates M as given by eq... HEOREM.: β,, ; ;, α β β γ / e tr β, ; ;, d. Φ α β γ > β for Re β > /. ROO: aking the M-transform of the right side of eq.. with resect to the variables and and the arameters and resectively, we have, + / + /, ; ;, Φ α β γ dd.5 > > which, under the transformation with d + / d and and then using the definition.5 yields, γ α β.6 α β γ Substituting this exression on the right side of eq.. and then integrating out in the resulting exression by using a Gamma integral gives, β γ α β β.7 β α β γ Now, taking the M-transform of the left side of eq.. with resect to the variables and and the arameters and resectively, we get,

6 6 > > + / α, β, β ; γ;, which, under the transformation + + / β dd / d and.8 with d and then using the definition. leads us to the same result as in eq..7. HEOREM.: β,, ;, ;, α β β γ γ / e tr β, ;, ;, d α β γ γ > β for Re β > /..9 ROO: aking the M-transform of the right side of eq..9 with resect to the variables and and the arameters and resectively, we get, / + / > >, ;, ;, α β γ γ dd. he alication of the same transformation to this exression as we have alied to the exression.5 above and then the use of definition.6 leads us to, γ γ α β. α β γ γ Substituting this exression on the right side of eq..9 and then integrating out in the resulting exression by using a Gamma integral gives,

7 7 β γ γ α β β. β α β γ γ Now, taking the M-transform of the left side of eq..9 with resect to the variables and and the arameters and resectively, we have, + / + / β > >,, ;, α β β γ γ ;, dd. which, on the alication of the same transformation as in eq..8 above and then using the definition. yields the same result as in eq.. above. HEOREM.: α, α, β, β ; γ + γ ;, γ + γ I / γ I γ + γ γ [, ; ; I I ]d α β γ for I and for Re, < < γ γ > /. I / α, β; γ; ]dd α, β; γ;. ROO: aking the M-transform of the right side of eq.. with resect to the variables and and the arameters and resectively, we get, + / + / > > [ α, β ; γ ; I Alying the transformations,, I + / I d and d I, with d I.5 + / d,

8 8 to the above exression and then alying eq...5 age 8 of Mathai [] leads us to, γ α β γ I α β γ α α β.6 β γ Substituting this exression on the right side of eq.. and then integrating out the variable in the resulting exression by using a tye - Beta integral results in M definition.. HEOREM.5: or, α [ α, α + /, β,α + / ; γ;, tr /, ; ;, e α Ξ α β γ d.7 > α where Re α > /. ROO: aking the M-transform of the right side of eq..7 with resect to the variables and and the arameters and resectively, we have, / / > > Ξ, ; ;, α β γ dd.8 Alying the transformation d and with d + to the above exression and then using the definition.8 roduces, γ α β.9 α β γ Substituting this exression on the right side of eq..7 and then integrating out in the resulting exression by using a Gamma integral leads us to,

9 9 γ α β α α. α α β γ Now taking the M-transform of the left side of eq..7 with resect to the variables and and the arameters and resectively, we get, / / α > > [, /,, / ; ;, dd. α α + β α + γ Making use of the transformation with d / d and + in the above exression and then using the definition. along with the observation that for, [ α + / ] [α + / ] α. [ / ] [ / ] α + α + α from eq. 6. age 8 of Mathai [], finally leads us to the same result as in eq.. above. It is to be noted that this result is different from the corresonding result in the scalar case. HEOREM.6: α, ;, α β γ γ ; α / ;, e tr α β γ γ ; > for Re α > /.,, d. ROO: aking the M-transform of the right side of eq.. with resect to the variables and and the arameters and resectively, we have,

10 > > / β; γ, γ ; which, under the transformations, + /, with / dd d / + d, d d d and, and then using the definition. leads us to the same result as in eq..6.. / + d and, and then using the definition.7 yields, γ γ β.5 β γ γ Substituting this exression on the right side of eq.. and then integrating out in the resulting exression by using a Gamma integral roduces, γ γ α β α.6 α β γ γ Now, taking the M-transform of the left side of eq.. with resect to the variables and and the arameters and resectively, we get, + / + / > > α, ;, ;, α β γ γ dd.7 which, under the transformations, with d + / d, References. Erdélyi A., Magnus W., Oberhettinger., ricomi. G. 95. ables of Integral ransforms, Vol. I, McGraw Hill, New ork, oronto and London.

11 . Exton H Multile Hyergeometric unctions and Alications, Ellis Horwood Limited, ublishers, Chichester.. Mathai A.M. 99. Jacobians of Matrix ransformations I, Centre for Mathematical Sciences, rivandrum, India.. Mathai A.M. 99. Hyergeometric unctions of Several Matrix Arguments, Centre for Mathematical Sciences, rivandrum, India. 5. Mathai A.M. 99. Aells and Humberts unctions of Matrix Arguments, Linear Algebra and its Alications; 8, Mathai A.M Secial unctions of Matrix Arguments-III; roceedings of the National Academy of Sciences, India; LV IV Saxena R.K., Sethi.L. & Guta O Aells unctions of Matrix Arguments. Indian J. ure Al. Math.; 8, no., Srivastava H.M., Karlsson.W Multile Gaussian Hyergeometric Series. Ellis Horwood Limited, ublishers, Chichester. 9. adhyaya Lalit Mohan, Dhami H.S.Nov.. Matrix Generalizations of Multile Hyergeometric unctions, # 88 IMA rerints Series, niversity of Minnesota, Minneaolis,.S.A.