SOME NEW IDENTITIES INVOLVING I-FUNCTION OF TWO VARIABLES

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Aryabhatta Journal of Mathematics and Informatics (Impact Factor- 5.856)

SOME NEW IDENTITIES INVOLVING

I-FUNCTION OF TWO VARIABLES

By

Raghunayak Mishra Department of Mathematics

Narayan Degree College Patti, Pratapgarh (U.P.)

&

Dr. S. S. Srivastava

Institute for Excellence in Higher Education Bhopal (M.P.)

ABSTRACT

Looking into the requirement and importance of various properties of identity in various field, in this paper we established some new identities involving I-function of two variables.

1. INTRODUCTION:

The I–function of two variables introduced by Sharma & Mishra [2], will be defined and

represented as follows:

I[yx] = I_p_i_,q_i_:r:p

i′,qi′:r′:pi′′,qi′′:r′′

0,n:m1,n1:m2,n2 _[

y x_|

[(bji:βji,Bji)_1,qi]

[(aj:αj,Aj)1,n],[(aji:αji,Aji)_{n +1,p i}]

]

:[(dj;δj)_{1,m 1}],[(d_ji′;δγ_ji′)_{m 1+1,q i′}];[(fj;Fj)_{1,m 2}],[(f_ji′′;F_ji′′)_{m 2+1,q i′′} ]

:[(cj;γj)1,n 1],[(cji′;γji′)n 1+1,pi′];[(ej;Ej)1,n 2],[(eji′′;Eji′′)n 2+1,pi′′ ]

= _{(2πω )}1 ₂ ϕ_L₁ _L₂ 1 ξ, η θ2 ξ θ3(η)xξyηdξdη, (1)

where

ϕ₁ ξ, η = nj=1Γ(1−aj+αjξ+Ajη)

[

r

i=1 p ij=n +1Γ(aji−αjiξ−Ajiη) j=1qi Γ(1−bji+βjiξ+Bjiη) ,

θ₂ ξ = j=1m 1Γ(dj−δjξ) j=1n 1 Γ(1−cj+γjξ)

[ q i′_{j=m 1+1}Γ(1−d_ji′+δ_ji′ξ) p i′_{j=n 1+1}Γ(c_ji′−γ_ji′ξ)

r′ i′ =1

,

θ₃ η = j =1m 2Γ(fj−Fjη) n 2j=1Γ(1−ej+Ejη)

[ qi′′_{j=m 2+1}Γ(1−f_ji′′+F_ji′′η) p i′′_{j=n 2+1}Γ(e_ji′′−E_ji′′η)

r′′ i′′ =1

,

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n2 0, qi >0, qi´  0, qi´´  0, (i = 1, …, r; i´ = 1, …, r´; i´´ = 1, …, r´´; k = 1, 2) also all the A’s, ’s, B’s, ’s, ’s, ’s, E’s and F’s are assumed to be positive quantities for standardization purpose; the definition of I-function of two variables given above will however, have a meaning even if some of these quantities are zero. The contour L1 is in the plane and runs from –  to + , with loops, if necessary, to ensure that the poles of djj) (j = 1, ..., m1) lie to the right, and the poles of cj  j) (j = 1, ..., n1), aj + j+ Aj) (j = 1, ..., n) to the left of the contour.

The contour L2 is in the plane and runs from –  to + , with loops, if necessary, to ensure that the poles of fj  Fj) (j=1,..., n2) lie to the right, and the poles of ej  Ej) (j = 1, ..., m2), aj + j+ Aj) (j = 1, ..., n) to the left of the contour. Also

𝑅′ ₌ _α

ji pi

j=1 + γji′

p_i′

j=1 − βji

qi

j=1 − δji′

q_i′

j=1 < 0,

𝑆′ ₌ _A

ji pi

j=1 + p_j=1i′′ Eji′′ − _j=1qi B_ji − _j=1qi′′ Fδ_ji′ < 0,

𝑈 = pi α_ji

j=n+1 − qj=1i βji + j=1m1 δj − δji′

q_i′

j=m1+1 + γj

n1

j=1 − γji′

p_i′

j=n1+1 > 0,

(2)

𝑉 = − p_j=n+1i Aji − q_j=1i Bji − mj=12 Fj − Fji′′ q_i′′

j=m21+1 + Ej

n2

j=1 − Eji′′

p_i′′

j=n2+1 > 0,

(3) and |arg x| < ½ U, |arg y| < ½ V

In the present investigation we require the following formula:

From Rainvile [1]:

1

Γ α Γ(1−α) =

sin πα

π =

eiπα−e−iπα

2πi , (4)

Γ 1₂− z Γ 1₂+ z = π secπz, (5)

Γ z Γ 1 − z = π cosecπz, (6)

Γ α − r Γ 1 − α + r = (−1)r_{Γ 1 − α Γ α}_. ₍₇₎

2. IDENTITIES INVOLVING I-FUNCTION OF TWO VARIABLES:

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Aryabhatta Journal of Mathematics and Informatics (Impact Factor- 5.856) I_p

i,qi:r ∶p_i′+1,q_i′+1:r′∶p_i′′,q_i′′:r′′

0, n ∶m1,n1 ∶m2,n2 𝑧1

𝑧₂|………..,,∶………., σ,μ :………,……..………..,:………, σ,μ :………,……..

= 2πi −1{eiπσ I_p

i,qi:r ∶p_i′,q_i′:r′∶p_i′′,q_i′′:r′′

0, n ∶m1,n1 ∶m2,n2 _𝑧1e−iπμ

𝑧₂ |………..,,∶……….:………,……..………..,:………:………,……..

−e−iπσI_p

i,qi:r ∶p_i′,q_i′:r′∶p_i′′,q_i′′:r′′

0, n ∶m1,n1 ∶m2,n2 _𝑧1eiπμ

𝑧₂ |………..,,∶……….:………,……..………..,:………:………,…….. }, (8)

|arg 𝑧1| < ½ U, |arg 𝑧2| < ½ V, where U and V is given in (2) and (3) respectively;

eiπσ I_p

i,qi:r ∶pi′,qi′:r′∶pi′′,qi′′:r′′

0, n ∶m1,n1 ∶m2,n2 _𝑧1eiπμ

𝑧₂ |………..,,∶……….:………,……..………..,:………:………,……..

= 𝜋{I_p

i,qi:r ∶p_i′+1,q_i′+1:r′∶p_i′′,q_i′′:r′′

0, n ∶m1,n1 ∶m2,n2 𝑧1

𝑧₂|………..,,∶………., 1₂−σ,μ :………,…….. ………..,:………, 1₂−σ,μ :………,……..

+ 𝑖 I_p

i,qi:r ∶pi′+1,qi′+1:r′∶pi′′,qi′′:r′′

0, n ∶m1,n1 ∶m2,n2 𝑧1

𝑧₂|………..,,∶………., 1−σ,μ :………,……..

………..,:………, 1−σ,μ :………,…….. _}

, (9) |arg 𝑧1| < ½ U, |arg 𝑧2| < ½ V, where U and V is given in (2) and (3) respectively;

I_p

i,qi:r ∶p_i′+1,q_i′+1:r′∶p_i′′,q_i′′:r′′

0, n ∶m1,n1 ∶m2,n2 𝑧1

𝑧₂|………..,,∶………., σ+r,μ :………,……..………..,:………, σ+r,μ :………,……..

= (−1)r I_p

i,qi:r ∶p_i′+1,q_i′+1:r′∶p_i′′,q_i′′:r′′

0, n ∶m1,n1 ∶m2,n2 𝑧1

𝑧2|………..,,∶………., σ,μ :………,……..

………..,:………, σ,μ :………,…….. _,

(10)

|arg 𝑧1| < ½ U, |arg 𝑧2| < ½ V, where U and V is given in (2) and (3) respectively.

Proof:

To establish (8), replace I-function of two variables by its equivalent contour integral as given in (1), after using (4) and finally interpreting by (1), we obtain (8).

Proceeding on the same lines the results (9) and (10) can be easily established.

3. PARTICULAR CASES:

I. On specializing the parameters in main result, we get following identities in terms of I-function of one variable:

I_pm,n_i_+1,q_i_+1:r[z|_{…….,…….., σ,μ}……..,……., σ,μ ]

= (2πi)−1{eiπσIpi,qi:r

m,n _[ze_−iπμ_|

…….,……..

……..,……._]_−e−iπσ_I pi,qi:r

m,n _[ze_iπμ_|

…….,……..

……..,……._]}_{, (11)}

provided that |argz| < ½ B, where B is given by 𝐵 = 𝑛_{𝑗 =1}𝛼_𝑗 − _{𝑗 =𝑛+1}𝑝𝑖 𝛼_𝑗𝑖 + 𝑚_{𝑗 =1}𝛽_𝑗 −

𝛽𝑗𝑖 𝑞𝑖

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= 𝜋{I_pm,n_i_+1,q_i_+1:r[z|

…….,…….., 1₂−σ,μ ……..,……., 1₂−σ,μ

]+𝑖I_pm,n_i_+1,q_i_+1:r[z|_{…….,…….., 1−σ,μ}……..,……., 1−σ,μ ]}, (12)

𝛽𝑗𝑖 𝑞𝑖

𝑗 =𝑚 +1 > 0;

I_pm,n_i_+1,q_i_+1:r[z|_{…….,…….., σ+r,μ}……..,……., σ+r,μ ]

= (−1)rI_pm,n_i_+1,q_i_+1:r[z|_{…….,…….., σ,μ}……..,……., σ,μ ], (13)

𝛽_𝑗𝑖

𝑞𝑖

𝑗 =𝑚 +1 > 0;

II. On choosing r = 1 in the integrals (11) to (13), we get following identities in terms of H-function of one variable:

H_p+1,q+1m,n [z|_(b

j,βj)1,q, σ,μ

(aj,αj)1,p, σ,μ _]

= (2πi)−1{eiπσH_p,qm,n[ze−iπμ|_(b j,βj)1,q

(aj,αj)1,p

] − e−iπσ_H

p,qm,n[zeiπμ|(bj,βj)1,q

(aj,αj)1,p

] }, (14)

provided that |argz| < ½ A, where A is given by 𝐴 = 𝑛_{𝑗 =1}𝛼_𝑗 − 𝑝_{𝑗 =𝑛+1}𝛼_𝑗 + 𝑚_{𝑗 =1}𝛽_𝑗 −

𝛽_𝑗

𝑞

𝑗 =𝑚 +1 > 0;

eiπσ H_p,qm,n[zeiπμ|_(b j,βj)1,q

(aj,αj)1,p ]

= 𝜋{H_p+1,q+1m,n [z|

(bj,βj)1,q, 1₂−σ,μ

(aj,αj)1,p, 1₂−σ,μ

]+𝑖H_p+1,q+1m,n [z|_(b

j,βj)1,q, 1−σ,μ

(aj,αj)1,p, 1−σ,μ _{] }}

; (15)

𝛽𝑗 𝑞

𝑗 =𝑚 +1 > 0;

H_p+1,q+1m,n [z|_(b

j,βj)1,q, σ+r,μ

(aj,αj)1,p, σ+r,μ _]

= (−1)rH_p+1,q+1m,n [z|_(b

j,βj)1,q, σ,μ

(aj,αj)1,p, σ,μ _]

, (16)

𝛽_𝑗

𝑞

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Aryabhatta Journal of Mathematics and Informatics (Impact Factor- 5.856) REFERENCES

1. Rainville, E. D.: Special Functions, Macmillan, NewYork, 1960.