Vol 2, No 7 (2011)

(1)

Available online through

_{www.ijma.info}

FORMATION OF CERTAIN SUMMATION FORMULAE

USING HYPERGEOMETRIC FUNCTION

Salah Uddin*

P. D. M. College of Engineering, Bahadurgarh, Haryana , India

Emails:

[email protected]

;

[email protected]

(Received on: 01-07-11; Accepted on: 14-07-11)

---

ABSTRACT

The main objective of the present paper is to establish certain new results in the aspects of Hypergeometric function.

Key words

: Gaussian Hypergeometric function, Recurrence relation, Bailey summation theorem.

2000 MSC NO

: 33C05, 33C20, 33C60, 33C70.

---

I. INTRODUCTION

Generalized Gaussian Hypergeometric function of one variable is defined by:

A

F

B

(

,a

2

,…,a

A

;b

1

,b

2

,…b

B

;z ) =

(1)

or

A

F

B

((a

A

);(b

B

);z)

A

F

B

((a

j

)

A

j=1

;(bj)

B

j=1

; z) =

(2)

where the parameters b

1

, b

2

, ….,b

B

are neither zero nor negative integers and A , B are non negative integers.

Contiguous Relation is defined by:

[Andrews p.363 (9.16), E.D. p.51 (10), H.T.F.I. p.103 (32)]

(a-b)

2

F

1

(a, b ; c; z) = a

2

F

1

(a+1,b; c; z) - b

2

F

1

(a,b+1; c; z) (3)

Recurrence relation is defined by

( +1)

=

( ) (4)

Bailey summation theorem[Prud, p.491(7.3,7.8)

2

F

1

(a, 1-a ; ; ) =

_!

=

_$! "# _!

(5)

II. MAIN SUMMATION FORMULAE

2

F

1

(a, -a-27 ; ; ) =

"#_{$ %}

[

& ' '( ')* ** & + , - _/,(,(&-*_! .' '( ')* ** &

+

& + , - ,(,(&-* . --( - **&) '()_/ + * (*(&)*-,-- (_! 0+ ( )(&-- '*&1

+

& ,- ,*) ) '- 2. ))- )(&)* (,3+ , (*-,)* &_/ _!%+ - '( * )4+ )-',- 5. ,*( -, 6

+

& &)& . - .)'' , ,* *''* 7+),*)((, )) *( ,* 7 . '_/ _! )''**','&& 7

+

& +-) (' *) &('* (07+ &- )) ,,&) -_/17 . &( (,'- &',,_! 27 . ,&,& &),- ))37

(2)

Page: 1122-1126

+

& ' ( ()-&%7+, & *( ,,47 8 (-),, 57 8 ' --)_/ _!67 . '& 7 . ) 7.*'',' * &(, &,* 7

+

& + &* )( ,) , (- (( 7 . ( ** ( *,-, ''*_/7 +&'(_! ((,)* 07 +, ,& ( (, ( 17

+

& + ) '(* *&-* 27 . )( - &(-- 37 . &-)((&&(_/ _!%7 + &*',, 47 + )-'* 57 +()'* 67

+

& -&(' * ' &(-& (70+ ,&*' '&,**-' &,* 7_/0. )*)'( ) ) (- ( 7_! 0. ,'(-('--,* * 070

+

& +(' )- ) *( 170+ ') ( & ( 270_/. )*, &(* (_! 370. ' -',, %70. *'-*( 470

+

& +' & 570+ & 670. (, &,(- '))-' &7_/1+' _!-*-)- -)(,* 71. ''( (, '( & 71

+

& & ' (, & ( & 071+* ( '*,* 171+' & (( (_/ _! 271.&,*(- ( 371. & , ' %71

+

& ' & 471. *& ,,* - * ,,'*72+&'- & **,)**& 7 / ! 2. '* )*** )) ( 72. *-, ), ' ' 072

+

& + ,&&,'*& ,*172+ -) &'(& 272+ )&*)-*(_/372. )) (_! %72. * 472. ,&- & , ( 73

+

& +& ) '&,-- && 73. ,- *,*&&( 73._/(-)))& (_! 073. ( *-- ) 173+,** - ( 273

9

& +*-((( 373. ,-) * *,**&7%+ ,),-)) &'-*( 7 / ! %+'&-(, ) -&& 7%. &' &&- ( 07%

+

&:** (*& 17%+* ( 27%+)*( 37%. &'-(& (& , 7_/ _! 4+ -, ** (*& 74+ (,&()&(( 74;

+

& ((-,)&& 074. *&)'* 174. *--(''*) *(7 /5+ -- (,-! 75+ ,&, ( 75. )* ' 075

+

& & ( 175. '* &('(& 76+&)&&'-*( 7_/6+ ),) (& 7_! 6. , , , (7 +'' ))* 7

+

& + (( 7 . *-_/ &7 . & -*7_! 0

+

+& )((( (,)*,', &_<.', , ,*&-( -&)_!

+

+& )((( (,)*,', & .', , ,*&-( -&)_< +)&' '( '-',' (*)_! 0.''),((, -*, ( 1

+

),(*( --) * )* 2. ,&( ) * ,)*'&3+ ('&&*&-(--,_< _! %+-&- &' )*, 4. &*,** & '5

9

''(,&) 6. (,', ' . '(, + (- 0 <+ 1.& )((( - -- ')! 7 + &* ()*)--& -* 7

+

&(, - ((&' &&)''&& 7+,) & -(-)&) ' ),_<07 + '_! ) ' &', (&(17 . &&)'') ',-* 27

+

( (, ''& 37 + ) ('*'&)( &%7 + ,'',, ( )(_< _! 47 + ,-- ' * 57 . ,--')-* 67

+

& ))* 7 . ,&( 7 . **&&) '' * *, ( 7 +(,*--)- ) (&')_< _! 7 . * ))) '- )& ( *& 7

+

+ &' -&)&(( &' & 07 + -) (&* &) -_<17 .) ,)&& (,&***_! 27 . ), ( &'() ( 37

+

'-&*--& )*& %7 +) *& &'& 47 + ** & '& < ! 57 +&&&( 67 . ,()* 7 .-( 7

+

&&* ((**-'-'-* -*70+')'*))& & () &(*& 7_<0. )' (' -'&-(,, '_! 70+**)'', ,)-) 070

+

+&)(* ('(,(-&& 170+ -& &(& ' 270._<,& &&*'()_! 370. ' &-& (& %70+ ' '))* 470

+

+ )'* 570+( ,'* 670. ) & & )' ' '*(7 < 1+-( (,))! )), & 71. , & &* ( -* 71

+

''( ' - (- ( * 071+,)((** ( * 171+ * '**- ) &_< _! 271. ,'-*& &( -*371. * & * ( %71

+

&' ) 471+ *( 571+ ,*( 671. & )*,*'**-&* ,( &7_< _! 2+ *(* ,, ()-*( 72

+

(()*- ,& ) 72. )*,,- ',&(- (_<072+',& &(&' &*&_! 172+ )-- & - (& 272

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Page: 1122-1126

+

*&,(,,& * )'&& 73. ))*- ) ,& &073_<+)& ''' ,')*_! 173+- ), ,)- 273+( **' &373

+

* *&%73. -& ( 473. ( -'*& )* _<(7%+ ,(( -)', )'&) 7_! %.- ,-( , (& 7%

+

&&( ) (,* 07%. ' *&* && 17%+ *&')_< _!27%+ )-,, 37%.) )-( ',* &74

+

+((- && '-&), 74+ ( ' ( -))* 74_<. , ')- '*_! 074. ' ) & 174+ )) ( 274

+

+ *(( 374. &'*-(&-, * ' 75+''(&)' ), & 7_<5+()--(' &_! 75. )*),*&( 075. , , 175

+

-' * , ,*76+('* )&,- 76+ ( (-* <76! . ((-*076. '-& &176.,-*-)'-))*7

+

+ )**&*& 7 +&&) (' 7 . ,)& (7 +))& && 7 + (*)_< _! 7 . &- -&&70. ( - 71

] (6)

2

F

1

(a, -a-28 ; ; ) =

"#_{$ 4}

[

+(',))*,))),)) * . )' ,*

< !

+

+ ),& )---,-' -,*(0.*',&)&'(&,')'&&(_< 1. &',()& ** *-&_! 2+'&(&)*)**, &*3

+

+,) &&&) (, )- %+,*&'-(( ,( 4_<. *'*),&,'&,_! 5. - & ) 6. )-((&)

+

+' '*) +'(, 0+ 1. (',))*,('-(&)& & 7+ * , -_< _! * ')&& 7. &,*,*)* *' ,((( 7

+

+ &- -( &' &,& ' 07+ & '**('*-& ,*_<17. , -'--& -(**&_! 27. *(*' &&- )&( 37

+

+ '', '' * *,(%7+)*&)* (- * 47.&&) ,*--_< _!57 . &,)& &, 67 . ,&&( 7 . &)* 7

+

+ & 07 . ' (-& **' & ' ,* 7 + )* *) **)-,& &&&( 7 . ,(_< _! '()( &(*),(& 7

+

+&,-,(*, ( ),)( & 07 + -)(, )) '&,)-(&_<17 . & '*(&-,)& &_! 27 . &-'* , *- (37

+

+ & '' &)'* %7 + ,'*&(-** &47 + ,) * *_< _!57 . --, ,* 67 . '*-* 7 . 7

+

(- &'))''()()- - 70+)))(*()( (,'' )'* 7_<0. ) ')) * *&' ( '(& 7_! 0+,)& & *&&)* ,)*070

+

+- (&))*()& 170. )* ) &*& <270. & (( - - &! 370. &)(,)', & %70

+

+ *&( ( 470+ )- ,, 570+'-()_< 67_!0.&&( 70. '& ( ( '&* * )'*71

+

+ ) ) , -*- & ) 71. -( ,(-&' ' ,)* 7_< 1.' *)(& , -(&_! 071+ (-&,& , &(' 171

+

+ )&-* *( 271. ,,, )*& -- 3 <71.' ! &-&) %71+ &*(' 471+,,&& 571

+

+ * 671. (',' )''(- *&(72+'* (-,)(& -'))- 7_< _! 2. ' &' ( (,- ' 72

+

-)() -- ) , 072+-(,)-(&*(' (172+ &( &&-**(&(_< _! 272. &( *-'- ( 372. ,'')--*( %72

+

,( * ( 472+ & ' 572. '-- *-*'*(,-&-&&7_<3+& ('-*, *' ((*& 7_! 3. *,,&& & ' & & 73

+

-*-( , -&& 073+'('( & ( ) ( 173_<+ *&), ( (_! 273+ -('*( 373. -,*( %73

+

'&& 473. *,() ( , ,*7%+',')( '&'*( ,* 7_< _!%. )))- ,-** ( 7%.(*' ', , *((07%

+

+& **)))* 17%+, )- - 27%+' ,**&_< _!37%. ,'* %7%. & ,-* &) &&(74

+

+ '- '*- '(& 74. ',*-*() -'* 74. ' - * ,&&_< _!074. -(*,&) 174+*-*-* 274+& & 374

(4)

Page: 1122-1126

+

'(*, & , 76+ -- )&(( 76+& * -&- <76. )&'& &! 076. *),(&176. -'-&)-,, 7

+

+'&, (( 7 +),-'-(& 7 . &,)* 07 . &&, &( *7 + - (*& 7 +,' &( 7 . -( (((7_< _! 0

+

+( - 7_<0. *'(&7_! 1

+

'&-) -**'*),'- +,& ,-*& (&**)* &₌ _! . * & (( ),(*

+

+ )'-(*&&, ))&&, 0+ *' * '(-''& *₌1. (),&,,) -, )&(_! 2. (), ' &', &3

+

+'' ) -) %+' -&' -- ) '4+&' ,'- & = ! 5. - , *) 6. & &('' .')&

-+

+) 0+ 1. ( &(*&' ) -''((( 7+('( &) '---&' )( & 7 . *&( - )) ) * ,)* 7₌ _!

+

+ -&&*( ''' ' & 07+ (-( ) &-,*)&) *₌17 . ,*(,,---) ))&-*_! 27 . )-) ,',*&- 37

+

*'( &' (**- %7+-')(-(- *'*47+ *)--,)*' =57+-,'' )*! 67 . &) ,& 7 . ,,) 7 . & 07

+

) ',&') ,)' *(( 7 +, ()) &))*--( ,&,- 7 . -,-&&(*-' -( *'-*( 7 + )-,,)'**--)₌ _! 07

+

+) '( )*'))*- & 17 +,) &-&),,-'&&₌27 . ) ')- ,(,()_! 37 . &)- *(- ' '* %7

+

+, *)-*)(& 47 +'') (* ( 57 +',-' -* ₌67 .*)_! 7 . 7 . '&')* &*(( ( *' ) &70

+

+ (&- &' '&, )'-,&,* 70. ,*''* ,(, ' *,* 7₌0. )-( ),- * & -,'*_! 070+ -'* (-,**-')* 170

+

+& '))*'*((,)* 270. ')*-' * &' 370₌. *-, *-'&''*_! %70. &)((,')* 470+ ,**& 570

+

+ ))(,* 670+&&( 70. '), (*) ( '& &7₌1+& ,* ('' -'-,-*( 7_! 1. )* - )&) , ''* 71

+

,)&)( )(*,* 071+-(**( (, ) & 171+* '-')&- &&₌ _! 271. *-)& *&, 371. & &* * & %71

+

,'- 471+ ( 571+ * 672.₌&,&&',-'() --(&7_! 2+,-'- *& ) (* (&( 72

+

() &' (&' *(- ( 72. ( **) '*,- ₌(072+ * &)) ,*(('_! 172+&& -&'))& ) 272

+

+, *, &(- ( 372.- &),'- %72. ,-**)₌ 47_!2. & ' 572. ' '(-,' ,,,, * &&73

+

+,--', ()' ',* * 73. ', ''-& -&,- ₌73. _!&' )- ',,(&073. -& ' - &) 173

+

+ * (&()( ( 273+ ()&( * 373. ,')*₌ %7_!3. '&& 473. ' ,**&*-(, ') 7%

+

+& ,---)--* '*( 7%+) ',*&-'- *& 7%. -*(***(*))₌ _! 07%. - -' )* 17%+ ,(&-'&& 27%

+

+* *) 37%+ ,'* %7%. '-)& ( ' ) ) 7₌ 4+ , ((&')**)(& 7_! 4+*,'-*-*' )&& 74

+

,&'''')* 074. *))*,*' 174+ *) 27₌4+& &_!374. '' ),) (), 75+**-&-& *)- & 75

+

+ (-(& )- 75. ' ,- ) & 075. -, &₌175. ( *_! 275. ,'&(*,) ''- 76+ ',*&& & - 76

+

+) (,,-'* 7 6. ', *(076.*),(& 17₌6. , _! ('(&7 + ')(,&-)* 7 +-*,('*( 7

+

+ &,)*07 . (&)) ' &7 +,' &( 7 +,' &( 7 . '₌ _! *&7 0. ( - 7 0. *'(&71

] (7)

III Derivations of formula (5):

Putting b= -a-27, z= in known formula (2), we get

(2a+27)

2

F

1

(a, -a-27 ; c; ) = a

2

F

1

(a+1,-a-27; c; ) + (a+27)

2

F

1

(a,-a-26; c; )

(5)

Page: 1122-1126

REFERENCES

[1] Abramowitz, Milton. A and Stegun, Irene;

Handbook of Mathematical Functions with Formulas, Graphs, and

Mathematical Tables

. National Bureau of Standards, 1970.

[2] Garg, O. P, Salahuddin , Shakeeluddin; On Certain Summation Formulae Involving Hypergeometric Function,

International Journal of Computational Science and Mathematics.

, 2(2010), 67-76.

[3]

Salahuddin.; Certain new summation formula of Hypergeometric function,

Elixir, Dis Math..

, 35(2011), 2921-2924.

[4]

Salahuddin; Evaluation of a Summation Formula Involving Recurrence Relation,

Gen. Math. Notes,

2(2010), 42-59.

[5] Salahuddin; Two Summation Formulae Based on Half Argument Associated to Hypergeometric function,

Global

Journal of Science Frontier Research

. 10(2010), 08-19

.

[6] Salahuddin; Evaluation of Certain Summation Formulae Involving Gauss Theorem,

Global Journal of

Mathematical Sciences: Theory and Practical

. 10(2010), 309-316.

[7] Salahuddin; A Summation Formula Related To Bailey Theorem, Global

Journal of Science Frontier Research

,

11(2011), 53-67.