International Conference on Challenges and Applications of Mathematics in Science and Technology (CAMIST) January 11-13, 2010 On Symmetry Techniques and Canonical Equations for Basic Analogue of Fox's H-Function S.K. Sharma and Renu Jain 1 Department of Mathematics, Institute of Technology and Management, Gwalior ISchool of Mathematics and Allied Science, Jiwaji University, Gwalior ABSTRACT Having defined a basic analogue of Fox's H-function, we investigate its q-recurrence relations, q-differenceequation and the canonical equation, which associatedwith eachfamily of multivariable q-analogue of Fox's H-function is also obtained. INTRODUCTION We begin our study with q-hypergeometric functions rtps'...(1) Where a. = qui b. = q~j 1, J are complex variables (Pj 1=0,-1,-2,...) and we normally require that Ocq-cl. Note that, for n a nonnegative integer, we have Here ai'pj,x (a, qt = (1- a)(1- qa) -.. (1- qn-1a)...(2) Let Tu be the q-dilation operator corresponding function f of the variables u, v, w,... to the function. to the variable u, i.e. Tu maps a ± ±l Tu feu, v, w,...) = f(q u, v, w,...)...(3)
738 Challenges and Application of Mathematics in Science and Technology from (1.1), Agrawal et al. [1] introduce the recurrence relations. (1- a'tj,~,(:;:xj = (I-a, ),~t;~';x J I,; k,; r (4) (l-b,q-'tj~,( :;:xj = ~- b,q-'),~,(e~~/xj I,;I,; s (5) x-'(l-t) n. (a;. xj = (l-a,) (I-ar) n. (qa;. xj (6) x r or " b,' (I-b,) (I-bJ rors qb j ' where a if i=t=k e k a; = { q~:if i = k bj,ifj=t=l e b - I j - { q-ib., if j = l (7)... Saxena, et al. [2] introduced a basic analogue of the H-function in terms of the Mellin- Barnes type contour integral in the following marmer : Where 0 ~ m ~ B ; 0 ~ n ~ A; and i=h, G(qU) = fi{(i-qu+o 0=0 )JI...(9) Also, aj, f3 j are all positive integers and aj,b j are complex numbers. The contour C is a line parallel to Re(ws) = 0, with indentations, if necessary, in such a marmer that all the poles of G(qb)-fJ)s );1 ~ j. ~ m, are to its right, and those of G(ql-aj+a)s), 1~ j ~ n are to the left of C, that is if I~g(z) - W 2w ii10g Iz I}k 1t where Iq I <I, log q = -w = -(WI +iw 2) w, Wv W2 are definite quantities, WI and W2 being real.
On Symmetry Techniques and Canonical Equations for... 739 The Fox's H-function has been studied in detail by several mathematicians for their theoretical and application point of view. The detailed account of various classical special functions expressible in terms of Fox's H-function along with their applications can be found in the research monograph by Mathai, et al. [3,4]. In the present paper, we shall establish q-recurrence relations, q-difference equation of the basic analogues of Fox's H-function. We also investigate the symmetry operators and canonical equations associated with the family of multivariable basic analogue of Fox's H- function. RECURRENCE RELATIONS l~j~n,(aa,aa)]...(10) (1- aj-it-uj) Hm,n[X' (ai,al) q x A,B,q (bi'~i) n+1~j~a,(aa,aa)]...(11) ( 1- bjt-~j)hm,n[x' (al,a[) q x A,B,q (b.,~[) _Hm,n[. (al,al), - A,B x,q (bl'~i)' (bj +l'~j)'" (bm'~m),(bm+i'~m+l)"" 1~ j~m,(aa,aa)]...(12) (1-q-bjT~j) Hm,n[x.q (ai,a}) x A,B' (b},~} ).
740 Challenges and Application of Mathematics in Science and Technology mn[ (al,al), =HA:B x;q (bl,a P I ), (b m'pm' A ) (b m+i'pm+1 A ),... (b +1 A ) j 'I-'j.., m+1~j~b,(aa,(la)]...(13) x(l-t-i) Hm,n[X' (al,al),(aa,aa)] x A,B,q (bl'~i) m n[ (a, +0.1,0.1)...,(aA +aa,aa)] =-HA'B x;q, (bl +~I'~I)...,(bB +~B'~B) Proof of (2.1) To prove the result, we consider I-a u m n [ (al,al)'",(aa,aa)] (l-q JTxJ)HA:B x;q (b A) (b A),1~j~n I'l-'l..., B'PB On making use of definition (1.8), (1.3) and (1.9), the above expression becomes man IT G(q bj-t'js) IT G(ql-aj+up )(1- ql-aj+ujs )1tXs =~ fj=lb j=1 A ds 2m C IT G(q I-bj+~p) IT G(q aj-ujs )G(q I-s ) sin 1tS j= m+i j=u+i l~j~n...,,(aa,(la)]... (14) This completes the proof of the result (10). Similarly, we can obtain the results (11), (12), (13) and (14). Note that, relations (2.1) to (2.5) imply the fundamental q-difference equation satisfied by the H~:~. x(1- r;' )(1- ql-al T:I )UI ".(1- ql-an T:n )un (l_qan+i-it;un+1 )Un+1.,,(1-qarlT;uA )ua +(1- q b l T;~I )~I ".(1- q bmt;~m )~m (1- q -bm+1t~m+1)~m+1.,,(1-q -bbt~b )~BH~:~= 0
On Symmetry Techniques and Canonical Equations for... 741 CANONICAL EQUATION Now, we define the basis function FA~; of A+B+l variables by...(15) The q-difference operators ~1are defined by ~1f(Uj)=ujl[f(Uj)-f(q±Uj)]...(16) In terms of operator (3.2) and with the help of the relations (10) to (14),we obtain the following recurrence relations. 131 13B I mn ua+l"'ua+b- =F '.' [ u l ",u A ua+b+l A,B U 1 UA ' q (a,.u.),..., (aj -l,a j)...,(an, an ),(an+l,an+l)...,(a A,aA)]. (bl'~l)'...,,...(17) 131 13B I =Fm,n ua+l",ua+b.--- A,B[ U 1 U A U u 1... UA A+B+l
742 Challenges and Application of Mathematics in Science and Technology q (ai,ai),..., (an,an),(an+i,an+l)...,(aj -l,aj)...,,(aa,ua)]. (bl,f)i)'...,,(bb,f)b), n+l~j~a..(18) ~! ~B 1 =Fm,n UA+I UA+B.--- A,B [ a! aa U u l...ua A+B+I A+m+l~j~A+B...(19),(aA,(bB +UA,UA)] +f)b,f)b)...(20) On making use of relations (17) to (20), equation (14) becomes the canonical q-difference equation A-a! A-an A+an+1 A+aA A+ A-~! A-~m A+~m+1 A+~B \""m,n 0 LlI,... Lln Lln+I '... LlA LlA+B+I +LlA+I'... LlA+mLlA+m+l LlA+Bft'A, B - (...(21) The q-difference operators ~1(n +1~ j ~A), Sj (A + 1~ j ~A + m), ~: (A + m + 1~ j ~ A + B) and ~:+B+I are symmetry operators for the canonical equation (21). The detail account of the q-difference equations, canonical equations and symmetry operators is available from the monograph due to Kalnins and Miller [5], Miller [6,7].
CONCLUSION On Symmetry Techniques and Canonical Equationsfor... 743 We conclude with the remark that besides the properties and results proved in this paper, are likely to have applications to a wide range of problems of mathematical and physical sciences. REFERENCES 1. Agrawal, AK., Kalnins, E.G. and Miller, W.; Canonical equations and symmetry techniques for q-series, SIAM, J. Math. Anal. 18, (1987), 1519-1538 2. Saxena, RK., Modi, G.c., Kalla, S.L.; A Basic analogue of Fox's Hsfunction, Rev. Tec. Ing. Univ., Zulia, 6, (1983), 139-143. 3. Mathai, AM., Saxena, RK.; Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences, Springer-Verlag, Berlin (1973). 4. Mathai, AM., Saxena, RK.; The H-function with Applications in Statistics and Other Discieplines, John Wiley and Sons, Inc., New York. (1978). 5. Kalnins, E.G. and Miller, W.; Symmetry techniques for q-series: Askey-Wilson polynomials Rocky Mtn. J. Math., to appear. (1989). 6. Miller, W.; Lie Theory and Special Functions, Academic Press, New York, (1968). 7. Miller, W.; Lie Theory and q-difference equations, SIAM J. Math. Anal., I, (1970), 171-188.