Representation of Five Dimensional Lie Algebra and Generating Relations for the Generalized Hypergeometric Functions

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1 Represenaion of Five Diensional Lie Algebra and Generaing Relaions for he Generalized Hypergeoeric Funcions V.S.BHAGAVAN Deparen of Maheaics FED-I K.L.Universiy Vaddeswara- Gunur Dis. A.P. India. E-ail : drvsb@luniversiy.in ABSTRACT The ai of presen paper is o derive he generaing funcions for he generalized hypergeoeric funcions by consrucing a five diensional Lie algebra wih he help of Weisner s ehod. Here he suiable inerpreaion o various paraeers have been given siulaneously. These generaing funcions in urn yield as special cases a nuber of linear generaing funcions o various classical orhogonal polynoials ; naely he Laguerre Herie (even and odd) Meiner Golieb and Krawchou polynoials. Many resuls obained as special cases are nown bu soe of he are believed o be new in he heory of special funcions. Subjec Classificaions () : 33C 33C4 33C8 Key words : Special Funcions Generalized Hypergeoeric Funcions Lie Algebra Generaing Funcions..INTRODUCTION DEFINITION AND NOTATIONS The ai of he presen paper is o derive a new class of generaing funcions for he generalized hypergeoeric funcions by consrucing a five diensional Lie algebra wih he help of Weisner s group-heoreic ehod [ [] [7] ]. In order o derive he eleens of Lie algebra a suiable inerpreaion o he inde are given. These generaing funcions in urn yield as special cases a nuber of linear generaing funcions o various classical orhogonal polynoials ; naely he Laguerre Herie (even and odd) Meiner Golieb and Krawchou polynoials. Many resuls obained as special cases are nown bu soe of he are believed o be new in he heory of special funcions. These orhogonal polynoials are of grea iporance and play an iporan role in differen branches of analysis such as haronic analysis quanu physics olecular cheisry nuber heory approiaion heory and he aheaical heory of echanical quadraures ec. Many auhors obained various ypes of generaing relaions(linear bilinear/ bilaeral) for differen orhogonal polynoials by he classical and Lie group-heoreic ehods [ [] [] [] [] [4] ] which are having any engineering applicaions[ [] [] ].I.K.Khanna and e al [7] sudied soe properies of he generalized hypergeoeric funcions as a funcion of four paraeers: () β ( γ)! ( ) F [ γ ; ; ] β Where is a non- negaive ineger is any finie cople variable such ha < and β independen of for if β dependen upon hen any properies which are valid for β independen of failed o be valid for β dependen upon. NOTATION: For sipliciy be denoed furher as if here is no change in any of he paraeers. Applicaions: The following special cases of have been obained. () + γ () li ( γ ) ψ = L ISSN : 49-93X Vol. No. 4 Nov -Jan 3 339

2 where ( γ ) ( ) L is he Laguerre Polynoials. H (3) ψ = ( ) () and li /! / (4) ψ = ( ) () H 3/ + li /! H ( ) and H ( ) where + are he even and odd Herie polynoials respecively. H μ ( ) ψ = ( ) () and ( μ+ )/ li /! H ( μ+ 3)/ + μ (6) ψ = ( ) () li /! μ μ where H ( ) and H + ( ) are he generalized even and odd Herie polynoials respecively. ( γ ) / ψ ρ = ρ M ( y γ ρ) y γ! provided.. ( ; ) λ λ ( 8) ψ ( e ) = e φ ( y; λ) y where φ ( y; λ ) is he Golieb polynoials. (7) ( ) ; M y γ ρ is he Meiner polynoials. ( N ) / (9) ( ) ψ P = P K ( y P N) y N! provided.. where ( ; ) ( ) ; K y P N is he Krawchou polynoials.. GROUP-THEORETIC DISCUSSION To apply he Weisner ehod in deail we need he following recurrence relaions which are saisfied by he funcion : d () ψ β γ ( ) = ψ β γ ( ) + ψ + β γ + ( ) d ( ) ISSN : 49-93X Vol. No. 4 Nov -Jan 3 34

3 and d ( + ) () ψβγ ( ) = ψ βγ + ( ) + d ( β ) ( β )( ) { ( β ) [( γ ) β + ( + ) ]( )} ψ ( ). + β γ+ The above wo independen differenial recurrence relaions deerine he linear ordinary differenial equaion d d () [4 ( ) ( β ) 4 ( ){[ γβ ( )]( ) ( β )} {4 ( ) d d ( ) ( β ) ( )[ γ β + ( ) ]}] ψ ( ) =. βγ Le us wrie he differenial equaion () in operaor funcional noaion as d by by y by z by and ( ) by u ( y z ) in (3) o ge d y γ z ψ β γ d (3) L( γ ) = 4 ( ) ( β ) D + {4( )[ γβ+ ( ) ]( ) ( β ) ]} D d + 4 ( ) + ( ) ( β ) ( ) d [ γβ + ( ) ] whered. d As in he Weisner group-heoreic ehod le us replace d by by y by z by and ( ) by u ( y z ) in (3) o ge d y γ z ψ βγ β β β u u u u (4) 4 ( ) ( ) 4 ( ) y + 4 ( ) z 4( )( ) + y z u u u u u ( ) y β( ) z + ( β z+ ) 4 ( ) ( β ) = y z where u ( y z ) = y z ψ ( ) is a soluionof(4). βγ Le L represens he parial differenial operaor of (4) i.e. u β β () L = L( y z ) 4 ( ) ( ) 4 ( ) y + 4 ( ) z y z y z 4( β )( ) + ( ) y β( ) z + ( β z + ) y z 4 ( ) ( β ). We now see linearly independen differenial operaors Ai ( i= 34) each of he for () () (3) (6) Ai = Aij ( yz ) + Aij ( yz ) + Aij ( yz ) y z such ha (4) () + Aij ( y z ) + Aij ( y z ); j = 34. ISSN : 49-93X Vol. No. 4 Nov -Jan 3 34

4 ψβγ = γ ψβγ (7) A [ y z ( )] a y z ( ) ψβγ = γ ψβγ A [ y z ( )] b y z ( ) 3 ψβγ = γ ψβγ A [ y z ( )] c y z ( ) + γ+ 4 ψβγ = γ ψ+ βγ + A [ y z ( )] d y z ( ) A [ y γ + ψβγ = γ ψ βγ + z ( )] e y z ( ). aγ bγ cγ dγ and eγ are funcions of γ and where independen of y z and. On he oher hand each A ( y z )( i j = 3 4; = 3 4) is an epression in y z and which is independen of γ. To find he group of operaors le us wrie (8) A= y A= z A3=. y z wih he help of (7) and recurrence relaions () and () we ge he following operaors: yz (9) A4 = yz ij which are such ha ( β ) y z z βy A = + y z y z ( β) ( β ) y z +. 3/ ( ) ψβγ = ψβγ [ ψβγ ( )] = γ ψβγ ( ) 3 [ ψβγ ( )] = ψβγ ( ) + γ+ 4 [ ψβγ ( )] = ψ+ βγ + ( ) γ + [ ψβγ ( )] = ( + ψ β γ + () A [ y z ( )] y z ( ) A y z y z A y z y z A y z y z A y z ) y z ( ). Clearly hese operaors Ai ( i= 34) saisfy he following couaion relaions: () [ A A ] = [ A A ] = [ A A ] = A [ A A ] = A [ A A ] = [ A A ] = A [ AA ] = ( A4 + A3 + ) [ AA ] = where[ AB ] = AB BA. The above couaor relaions show ha he se of { Ai : i= 3 4} generae a five diensional Lie algebra.i is clear ha he differenial operaor L can be epressed in ers of hese operaors as () L A = A4 4( ) ISSN : 49-93X Vol. No. 4 Nov -Jan 3 34

5 `which coues wih each Ai ( i= 34) ; ha is (3) L A 34. i = i = 4( ) Furher he Weisner group-heoreic ehod has been used o find he eended for of he group generaed by A i= 34 as follows: i ( ) = ( a ) ( ) = ( a ) ( ) = ( a ) (4) e f yz f e yz e f yz f ye z e 3 3f y z f y z e 3 / a yz a yz a yz = + < a ( β ) yz ( ) e f y z f y z ( ) = + ( ) e f y z f p q r s a( β ) a aβ z wherep= + q= y r= z+ yz z yz a a ( β ) yz a( β ) s = ( ) provided <. yz yz a yz γ Therefore we have f ( yz ) = y z ψ βγ ( ) in () we ge a ( β ) yz a ( β ) a4 yz where ξ = + ( ( a4a) + ) yz a a a a ( β ) a η = ye θ ze = + yz yz yz a ( β ) a ρ = 3 yz( ) yz () e e e e e f( y z ) = + f ( ξηθ ρ) a 4 yz + a( β ) yz a. yz a ( β ) ISSN : 49-93X Vol. No. 4 Nov -Jan 3 343

6 3.GENERATING FUNCTIONS γ Fro he above discussion i is clear ha u y z y z ( ) ( ) = ψ βγ is soluion of he syses: Lu= Lu= (6) ; ( A ) u= ( A γ ) u= Lu= Lu= ; ( A3 ) u= ( A+ A+ A3 γ ) u= Fro he resul (3) i is verified ha γ γ ( ψβγ ( )) = ( ψβγ ( )) S L y z L S y z 4( ) 4( ) γ Therefore he ransforaion ( ψ β γ ) By choosing i 4 () we ge a A a A a A a A a A where S = e e e e e S y z ( ) is also annulled by L. 4( ) γ = = = = and wriing f yz y z ( ) a i 3; a b a c γ ( ψβγ ) 4 (7) e e y z ( ) ( ) = ψ βγ in βγ γ ca ba c( β ) c = + yz yz c( β ) b{ yz + c( β ) }{ yz c} 3 3 yz( ) { yz( ) c( β )} ψ ( ζ ) γ y z γ. where ( ) c β ( bc) byz ζ = + +. yz Also we now ha (8) A γ + γ+ 4 y z ψ β γ ( ) = y z ψ ( ) + β γ+ and A γ γ + y z ψ β γ ( ) = ( + ) y z ψ β γ + ( ). Bu on he oherhand we have ca ba l γ b c l ( ψβγ ) = l e e y z l yz l!! (9) 4 l ( ) ( ) ( ) ( ) ( ) l= = ψ + l β γ+ l + l γ ( ) y z. ISSN : 49-93X Vol. No. 4 Nov -Jan 3 344

7 Equaing he epressions (7) and (9) we ge γ γ c( β ) c c( β ) (3) + yz yz 3 yz( ) b y z c y z c 3 { yz( ) c( β )} { + ( β ) }{ } ψ βγ ( ζ ) l b c = l!! l= = l l ( ) ( ) l ( l) ( yz ) ψ + l β γ + l + l ( ) c c( β ) where < < and <. yz yz DEDUCTIONS: The following generaing relaions have been deduced by giving differen values o b and c. I. When b= c= and replacing y z - by ω we have l ( ) ( ) l ω (3) ψ+ l β γ+ l l ( ) = ψ β γ ( ω ) l= l! + / ω where II. When b= c= and replacing y - z - by ω we have (3) γ γ ( + ) ω( β ) ω ψ β γ + ( ) ω = =! + <. / ω ( β ) ω( β ) ψ 3 β γ + ( ) where ω <. 4. APPLICATIONS βγ Furher using he condiions on ( ) we ge he following generaing funcions: ( ) ( γ + ) ( γ ). L ( ) L ( ).! ω = + ω = ψ as special cases enioned above ISSN : 49-93X Vol. No. 4 Nov -Jan 3 34

8 ( + ) ( γ ) ( γ ) ( γ ). L ( ) ( ) ep( ) L ( )! + ω = + ω ω + ω = ω <. = ( ) ( ) μ + K μ 3. H ( ) H ( ).! ω = + ω μ ( μ ) K μ + 4. ( ) H =! ( ) ω = ( + ω) ep( ω ) H [ ( + ω)] <. = ( ) ( ) μ + K μ. H ( ) H ( ).! + ω = + + ω ( ) μ ( μ+ ) K μ H + + H + = 6.! ( ) ω = ( + ω) ep( ω ) [ ( + ω)] <. ( ) ( ) ( y) ω 7. M ( y ; γ + ρ) = M [ y; γ ρ ( ω ρ )]. =!( γ ) ρ ( ) ( ) ( y) ω 8. M ( y ; γ + ρ) = ( + ω ρ) [ ω ρ( ρ )] =!( γ ) ρ M γ y γ [ y; γ ρ ( ω ρ ( ρ ))] ω ρ( ρ ) < provided < ρ < and y =... λ λ ( ) K φ λ ω = ω ( ) ( y) λ 9. e ( y ; ) e ep =! λ λ λ [ y;log( e e)] e <. φ ω ω y λ λ ( + ) ( + ). K e ( ; ) λ λ φ + y+ λ ω = ωe ( e ) ep =! λ λ λ [ y;log( e e λ ( e ))] e λ ( e ) <. φ ω ω ISSN : 49-93X Vol. No. 4 Nov -Jan 3 346

9 ( y). K ( y ; θ N )( ω θ ) = K [ y; θ( + ωθ θ ) N].!( N) ( ) = K K ( ) ( + N) ω () K + [ y ; θ N ] =! θ ( ) N ω y ωθ = + ωθ θ K [ y; θ (+ ω θ ) N] θ ω θ < provided < θ < y =... N. These are he well nown linear generaing relaions for he Laguerre Herie (even and odd) Meiner Golieb and Krawchou polynoials respecively in which soe of he are believed o be new in he heory of special funcions. Rear: In a siilar way one can easily be derived he corresponding generaing funcions for ψβγ ( ). Acnowledgeens: The auhor wish o epress sincere hans o Professor. I.K.Khanna Deparen of Maheaics Banaras Hindu Universiy Varanasi for his valuable suggesions which enabled he auhor o iprove he presenaion of he paper. REFERENCES [] G.Arfen Chebyshev Polynoials and Chebyshev Polynoials-Nuerical Applicaions in Maheaical ehods for Physiciss 3rd ed Orlando.F.L. Acadeic Press 98. [] E.B. Mc Bride Obaining Generaing Funcions Springer New Yor 97. [3] P.M.Cohn. Lie groups Universiy Press Cabridge 96 [4] M.E.H.Isail and D. Sanon Classical Orhogonal Polynoials as oens. Canad. J. Mah. 49 (3) [] I.K.Khanna and V. Srinivasa Bhagavan Weisner s ehod o obain generaing relaions for he generalized Polynoial se J. Phys. A: Mah. Gen [6] I.K.Khanna and V. Srinivasa Bhagavan Lie group-heoreic origins of cerain generaing funcions of he generalized hypergeoeric Polynoials Inergral Transfors and Special Funcions Vol. II No [7] I.K.KhannaV.Srinivasa Bhagavan.and M.N.Singh Generaing relaions of he hypergeoeric funcions by Lie groupheoreic ehod J. Maheaical physics Analysis and Geoery [8] A.M.Mahai and H.J.Haubold Special funcions for Applied Scien- issspringer New Yor 8. [9] A.M.Mahai H.J.Haubold and R.K.Saena The H-funcion: Theorey and Applicaions Springer Newyor 9. [] W.Jr.Miller Lie Theory and Special Funcions Acadeic Press New Yor 968. [] E.D.Rainville. Special Funcions Macillan Co. New Yor96. [] S.N.Singh and R.N.Bala Group-heoreic origins of cerain generaing funcions of he odified Lauguerre Polynoials Indian J.Pure Applied Mahs 7 No [3] I.N.Sneddon Special Funcions of Maheaical physics and Cheisry Iner Science New Yor 96. [4] H.M.Srivasava and H.L.Manocha A Treaise on Generaing Funcions Halsed/Wiley New Yor 984. [] H.M.SrivasavaSoe Clebsch-Gordan ype linearizaion relaions and oher polynoial epansions associaed wih a class of generalized uliple hypergeoeric series arising in physical and quanu cheical applicaions J. phys. A. Mah. Gen [6] C. A.Truesdell A Unified Theory of Special Funcions Princeon Universiy Press PrinceonNew Jersey948. [7] L.Weisner Group-heoreic origin of cerain generaing funcions Pacific J. Mah ISSN : 49-93X Vol. No. 4 Nov -Jan 3 347