On Hankel type transform of generalized Mathieu series

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1 Inernaional Journal of Saisika and Mahemaika ISSN: E-ISSN: Volume Issue 3 3- On Hankel ye ransform of generalized Mahieu series BBWahare MAEER s MIT ACSC Alandi Pune-45 Maharashra India Corresondence Addresses: balasahebwahare@gmailcom bbwahare@miacscacin Research Aricle Absrac: In his aer by using inegral reresenaions for several Mahieu ye series a number of inegral ransforms of Hankel ye are derived here for general families of Mahieu ye series These resuls generalize he corresonding ones on he Fourier ransforms of Mahieu ye series obained recenly by Elezovic eal [4] Tomovski [9] and Tomovski and Vu Kim Tuan [] Inroducion: Keywords: Inegral reresenaions Mahieu ye series Hankel ransform Fourier ransform Bessel ye funcion Fo-Wrigh funcion Sonine-Schafheilin formula Fo H-funcion Mahemaics Subec Classificaion: 44A 33A The infinie series S( r n ( r R ( n ( n r is named afer Emile Leonard Mahieu ( who invesigaed i in his 89 work [8] on elasiciy of solid sin( r bodies An inegral reresenaion of ( is given by (See [5] S( r d r ( Several ineresing resuls dealing wih inegral reresenaions and bounds for a sligh generalizaion of he Mahieu series wih a fracional ower defined as n S ( r (( r R ; α α > (3 α n n r can be found in he works by Diananda [] Tomovski and Trencevski [6] Cerone and Lenard [] and Tomovski [8] Srivasava and Tomovski [4] defined a family of generalized Mahieu series q ( q ( q a n Sα ( r : a S (( r:{ { ak} } k (( r q( ( α R (4 q ( a r n where i is acily assumed ha he osiive sequence ( n lim { n} n { 3 } ( n n a { a } { a a a } ( a is so chosen ha he infinie series in definiion (4 converges ha is ha he following auiliary series Coyrigh Saerson Publicaions Inernaional Journal of Saisika And Mahemaika ISSN: E-ISSN: Volume Issue 3

2 BB Wahare n ( q a α n is convergen Comaring he definiions ( (3 and (4 we see ha Furhermore he secial cases { } ( α α ( { } S ( r S( r and S ( r S ( r { k } r ( k { } S ( r :{ a } Sα ( r:{ k } and ( ( invesigaed by Qi [3] Diananda [] Tomovski [7] and Cerone-Lenard [] Definiions and formulas: ( { } (( α have been S ( r :{ k } In his secion we give some definiions and formulas needed for comuaion of he Hankel ransforms of ( q (( q r S( ( r Sα (( r Sα r : r and Sα (( r :{{ k }} In order o evaluae he Hankel ye ransform of S( ( r we firs ge α in Sonine-Schafheilin formula {see eamle [6 69] : λ J ( a J ( b d α a Γ( ( α 3 α / ( α ( λ α α α α λ 3α λ ( 3 (( 3 / α 3α λ 3α α λ a F ;3 α ; b ( R( α 3 α > R( λ > ; < a < b ( b Γ Γ ( wih a corresonding eression for he case when < b < a which is obained from ( by inerchanging a and ( α and( α In view of he relaionshi b and also J ( sin ( π we find from he Sonine-Schafheilin formula ( ha α a Γ ( ( ( ( α λ λ πb J (( a sin (( b d α λ 3α λ b Γ (( 3 α Γ (( λ ( ( α ( ( α λ α λ a F ;3 α ; b ( R ( ( α R ( λ a b ( ( ( > > ; < < (3 Inernaional Journal of Saisika and Mahemaika ISSN: E-ISSN: Volume Issue 3 Page 4

3 Inernaional Journal of Saisika and Mahemaika ISSN: E-ISSN: Volume Issue 3 3- ogeher wih he corresonding inegral derived from he analogue of ( for he case when < b < a Thus by furher alying inegral formula (3 and is comanion when < b < a we obain ( wih λ α α : π α where α (( Ω α ( ; F ;3 ; ( ( α α α α < < Γ ( 3 Γ ( ( Ω ( ( ; : ( α < < R α > α r sin( ( r Jα (( r dr (( (4 Ω (( ; : < < ( α ( α (( ( α α 3( ( α Γ Γ( ( α Γ ( 3 ( ; F α α ; ; ( ( R( ( α Ω 3 < < > If we aly he Sonine-Schafheilin formula ( firs wih λ α 3 α a b ( ( < a < b and hen wih λ α α α α a b ( b a 3 ( < < we ge where R ( α R ( α (( α 3 (( ; : r J (( r J (( r dr Θ α α < < α (( Θα (( α : :< < 3 ( > ( > (5 (( Γ (( α Θα (( α : 3 3( F α α : α : α α Γ( ( α Γ (( α 3 < < R( ( α > R( ( α > (6 α (( Γ (( α Θα (( α ; α 3 α 3α 5 Γ ( 3 α Γ( α α { α 3α 7 F α ;3 α ; 3 < < R( ( α > R (( α > (7 ( q For he evaluaion of he Hankel ye ransform of he general Mahieu ye series Sα r; r we need he definiion of he Meier G-funcion and he Fo H-funcion (for more deails see eg in [3 Vol [] Definiion : By a Fo s H-funcion we mean a generalized hyergeomeric funcion defined by means of he Mellin-Barnes-ye conour inegral Coyrigh Saerson Publicaions Inernaional Journal of Saisika And Mahemaika ISSN: E-ISSN: Volume Issue 3

4 BB Wahare H δ π i m n Γ( ( bk s qk Γ( ( a s k (8 l Γ( ( bk s qk Γ( ( a s m n au u u v bv qv q k m n where is a suiable conour in C he orders (( m n u v are inegers m q n and he arameers a R > b R q > k q are such ha ( b l q ( a l l l k k k k Many secial funcions are aricular cases of he H-funcion m n For eamle if l q ( ( l ; q i reduces o he Meier G-funcion Gu v (( δ (for definiion and roeries See [3 Vol] : m n (( au u m n (( au Η u v δ (( G bv qv u v δ (( bv (9 On he oher hand he Fo-Wrigh Ψ - funcion is he following secial case of he H-funcion see for eamle in [5]: µ ( ( au u u Ψ v ( au u ;( bv qv ; H u v (( (( bv qv ( Ne we use of he following Miller ransform of a roduc of wo hyergeomeric funcions roven by Miller and Srivasava [9] (see also [ Sec 333] s ( α ( α F( s r F ( ; ; a r F ( ; q 3 ; b r dr ( ( q b G Γ ( 3 α Γ ( 3 α Γ s/ 3 α s/ s 33 α q a Γ( a ( ( a > b > < R(( s < R ( 5α 3 α q < R(( s < R( 4α Subsiuing he relaion (See [ 77] Γ ( 3 ( F ( ;3 ; ( ( α a r ( α Jα ( ar α (( ar ino he inegral formula ( wih q α α s 4α 4α 3 a α q α q α α b we ge α q q r r Jα (( r F α ; α α ; dr 4 q q 3α α Γ α 3 Γ α α q α 3 η G 3 33 Γ( ( α (3 Inernaional Journal of Saisika and Mahemaika ISSN: E-ISSN: Volume Issue 3 Page 6

5 Inernaional Journal of Saisika and Mahemaika ISSN: E-ISSN: Volume Issue q R( ( α > R( ( α > R α > > > ( In order o evaluae he Hankel ye ransform ( q r Sα r { k } ( ;{ } we aly he known inegral formula from [ 3 355] wih q r : α ( α 3 J α ( σ H ω 4ω (( (( r ( q d H α r σ (4 3 ω σ R( ( α > R( ( α > R R Using relaion ( wih q by (4 we ge he following inegral formula: ( (( ( ( α ( α ( γ ( α q γ ( ( 3 ( 3 3 ( ( σ α γ Jα ( σ Ψ ( α ;( γ (( α q γ ; d γ 4 (( (( α 3 (( α H 3 3 (( (( γ (( α q γ σ σ 3 σ R( ( α > R( ( α > R R 3 Evaluaion of Hankel ye ransforms: The Hankel ye ransform of order α is defined by H ( f (( r (( f (( r J ( r ( r α d (3 α α where J α is he Bessel funcion of he firs kind and of order (( α wih R( ( α In view of he relaionshi ( and J (( cos (3 π The Hankel ye ransform reduces o he Sin-Fourier and Cos-Fourier ransforms: (( I s f (( sin( ( r f (( r dr π (( I c f (( cos( ( r f (( r dr π (5 For a generalizaion of he Hankel ye ransform as well as he Sin-Fourier and Cos-Fourier ransforms see Luchko and Kiryakova [7] Using he inegral formula (4 we firs evaluae he Hankel ye ransform of S(( r : ( H ( S( r ( S( r J ( r ( r α dr α α Coyrigh Saerson Publicaions Inernaional Journal of Saisika And Mahemaika ISSN: E-ISSN: Volume Issue 3

6 BB Wahare ( r sin 9 J (( r dr α e r ( ( α r sin( ( r J (( r dr d α π (( (( (( ; d (((( ; d α α Ω Ω (33 Using he relaion [6 4] 3 z F ; : z ln z z we ge (( 3 Ω (( ; F ; ; ( ln ( < < Γ( ( 3 / Γ( ( / π Hence (( 3 Ω (( ; F ; ; 3 ln (( < < Γ Γ π ( ( I s H ( S( r ( ( S( r ( π ln d ln d e π π ln d ln u π PV ln d (( π > (34 where he Cauchy Princial Value (PV of he las inegral is assumed o eis By a direc comuaion he same formula (34 was recenly roved by Elezovic e al [4] The inegral reresenaion of S3 (( r obained by Cerone and Lenard in [] is given by α ( r ( α α π S (( r J (( r d (( r α R (35 3α ( Γ ( 3 Alying inegral formula (5 we obain ( H ( S ( r ( S ( r J ( r ( r α dr α 3α 3α α α π α 3 r J (( r d (( 3 α Γ J α ( r dr α π α 3 r J (( r J (( r dr d α (( 3 α Γ Inernaional Journal of Saisika and Mahemaika ISSN: E-ISSN: Volume Issue 3 Page 8

7 Inernaional Journal of Saisika and Mahemaika ISSN: E-ISSN: Volume Issue 3 3- α π (( (( ; α α d Θ Γ (( 3 α α (( Θ (( ; d α α (36 α R( ( α > > R In order o evaluae he Hankel ye ransform of [4]: q α (( q S r; k α q Γ α q q r F α ; α α ; d 4 q r q R α > and inegral formula (3 Thus we have (( q (( q α H α Sα r ; r (( Sα r; r Jα (( r (( r dr ' ' q ( α q Γ α α q q r r F α ; α α ; Jα (( r dr d 4 (( q Sα r ; k we aly is inegral reresenaion from q α 3α α π 3 α G q 33 q q d α e α 3 Γ( ( α q q R( ( α > α > α > > (38 The inegral reresenaion of ( k ( α ( k ( q { } ( α (37 S ( r ;{ k γ } obained by Srivasava and Tomovski [4] is given by γ (( α q (( (( q γ γ Sα ( r;{ { k }} Ψ (( α ; (( γ( ( α q γ ; r d Γ( (39 Coyrigh Saerson Publicaions Inernaional Journal of Saisika And Mahemaika ISSN: E-ISSN: Volume Issue 3

8 BB Wahare ( r q y γ (( α q > ( R ( Using his inegral reresenaion and (5 we ge (( q γ ( α ( { } H ( S ( r;{ k } ( α (( q γ α Sα ( γ ;{{ k }} Jα ( y ( r dr Ψ ' ' y( α ( q α γ r J (( r (( ;(( ( q ; r dr d α α γ α γ Γ( α ( α γ (((( α q γ (( (((( α 3 (( α H 3 (( (( ((( q d γ α γ 4 e Γ 3 R( ( α > R( ( α > γ (((( α q > > (3 References: [] P Cerone CT Lenard On inegral forms of generalized Mahieu series J In- equal Pure Al Mah 4 No 5 Aricle - (elecronic 8 [] PH Diananda Some inequaliies relaed o an inequaliy of Mahieu Mah Ann [3] A Erdelyi W Magnus F Oberheinger and FG Tricomi (Ed-S Higher Transcedenal Funcions Vols --3 McGraw Hill N York-Torono-London 953 [4] N Elezovic HM Srivasava Z Tomovski Inegral reresenaions and inegral ransforms of some families of Mahieu ye series Inegral Transforms and Secial Funcions 9; No [5] O Emersleben Uber die Reihe Mah Ann [6] ISGradsheyn IM Ryzhik Table of Inegrals Series and Producs Academic Press N York-London 965 [7] YuFLuchko VSKiryakova Generalized Hankel ransforms for hyer-bessel differenial oeraors CR Acad Bulgare Sci 53 No8 7- [8] ELMahieu Traie de Physique Mahemaique VI- VII : Theory de l Elasicie des Cors Solides (Par Gauhier Villars Paris 89 [9] AR Miller and HM Srivasava On he Mellin ransform of a roduc of hyergeomeric funcions J Ausral Mah Soc Ser B [] TK Pogany HM Srivasava Z Tomovski some families of Mahieu a-series and alernaing Mahieu a- series Al Mah Comuaion [] AP Prudnikov Yu A Brychkov OI Marichev Inegrals and series Vol: Elemenary Funcions; Vol Secial Funcions Gordon and Breach Sci Publ NYork (99: Russian Original Nauka Moscow 98 [] AP Prudnikov Yu A Brychkov OI Marichev Inegrals and series (More Secial Funcions Gordon and Breach Sci Publ N York (99: Russian Original: Nauka Moscow 98 [3] F Qi An inegral eression and some inequaliies of Mahieu ye series Rosock Mah Kolloq [4] HM Srivasava Z Tomovski Some roblems and soluions involving Mahieu s series and is generalizaions J Inequal Pure Al Mah 5 No Aricle 45-3 (elecronic 4 [5] HM Srivasava and HL Manocha A Treaise of Generaing Funcions Halsed Press (Ellis Horwood Ld Chicheser J Wiley & Sons N York-Chicheser- Brisbane-Torono 984 [6] Z Tomovski ; K Trencevski On an oen roblem of Bai-Ni Guo and Feng Qi J Inequal Pure Al Mah 4 No Aricle 9-7 (elecronic 3 [7] Z Tomovski New double inequaliy for Mahieu Series Univ Beograd Publ Elekroehn Fak Ser Ma [8] Z Tomovski Inegral reresenaions of generalized Mahieu Series Via Miag-Leffler ye funcions Frac Calc and Al Anal No [9] Z Tomovski New inegral and series reresenaions of he generalized Mahieu Series Al Anal Discree Mah No 5-8 [] Z Tomovski Vu Kim Taun On Fourier ransforms and summaion formulas of generalized Mahieu Series Mah Science Res J; To aear 9 Inernaional Journal of Saisika and Mahemaika ISSN: E-ISSN: Volume Issue 3 Page

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