On certain continued fractions involving two basic hypergeometric series

IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 13, Issue 3 Ver. IV (May - June 2017), PP 64-67 www.iosrjournals.org

On certain continued fractions involving two basic hypergeometric series Dr. Jayprakash Yadav1, Vidhi Bhardwaj2 1

Prahladrai Dalmia Lions College of Commerce and Economics, Mumbai, India.

2 D.S.J. College, Mumbai, India

Abstract: The object of this paper is to establish continued fraction for the ratio of two 4 3 series by the application of some known transformation formula in basic hyper geometric series.

Keywords and phrases: Basic hyper geometric series, Continued fractions and some known results in basic hyper geometric series.

I. Definitions And Notations The basic hyper geometric series is defined as

(a); z   a ; q n z n ,  (b)  n 0 b ; q n q; q n

r s 

(1)

where for the convergence of the series, we have

0  q  1 and z  1if r = s+1, and for

any z if r  s.

a ; q n  a1 ; q n a 2 ; q n ....a r ; q n , b ; q n  b1 ; q n b2 ; q n ....bs ; q n .

(2) For real or complex, a, q <1, the q -shifted factorial is defined by (3)

We will also use the following results in our analysis:

q  n , a, d / b, d / c; q  q  n , a, b, c; q  e a, f a n n a 4 3    4 3  1 n 1 n e, f n d , e, f  d , aq / e, aq / f 

(4)

and 1 1  de  2 2  , q  ,  q  , a, b, c, dq n , eq n ;   aq f , bq f , cq f , abc f ; q  q n , a, b, c; q  abc  =    8 7 1 abq f , acq f , bcq f , q f ; q  4 3 d , e, f   12    ,  2 , q a , q b , q c , e, d 

(5) where

  abc f , def  abcq

DOI: 10.9790/5728-1303046467

1 n

[ Gasper and Rahman [2]; Appiii(iii.20) ]

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On Certain Continued Fractions Involving Two Basic Hypergeometric Series II. Main Results

q , a, b, c : q  q , aq, b, c : q   0  0 1  1  2  2 ( a ) 4 3  .......,  4 3   1   0  1  1  1   2  1  d , e, f  dq, e, f  dq 2i 1 (1  q i  n )(1  bq i )(1  cq i ) a2q2  Where  i = , = i aq  eaq  f 1  eq i 1  fqi  (1  dq 2i )(1  dq 2i 1 )(1  eq i )(1  fqi ) n

And

(b)

i=

n











(6)

ef 1  dq i  n 1 1  aq i 1 b  dq i 1 c  dq i 1 q i  2 n 1 . bc 1  dq 2i 1 1  dq 2i 1 e  aq 2 n i f  aq 2 n i 1  eq i 1  fqi









1 1   n n de 2 2   , q ,q , a, b, c, dq , eq ;  abc  8 7  1  12    ,  2 , q a , q b , q c , e, d 







 0  0 1  1  ( f  aq)( f  abcq )  .... 1  ( f  abq )( f  acq )   1   1    0 1   de n 1 n  , q ,q , aq, b, c, dq , eq ;  abc  8 7  1 1  2  2  , , q a , q b , q c , e, dq  1 2

=

1 2

(7)

  abc f , def  abcq 1n ,   abcq f , def  abcq 1n .

Where

,

and For i  o ,

Proof of (a):

q  n i , aq i , bq i , cq i : q  Let H i  4 3   2i i i dq , eq , fq  So,

H i  Fi 

This gives

Where

i 

 



and



q  n i , aq i 1 , bq i , cq i : q  Fi  4 3  2i 1 i  i dq , eq , fq 



dq 2i 1 1  q i  n 1  bq i 1  cq i H i 1 1  dq 2i 1  dq 2i 1 1  eq i 1  fqi









Hi i  1 , Fi Fi H i 1

 



(8)





dq 2i 1 1  q i  n 1  bq i 1  cq i .

1  dq 2i 1  dq 2i 1 1  eq i 1  fqi



Let us transform







Fi and H i by means of the transformation formula (4)

we get:

DOI: 10.9790/5728-1303046467

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On Certain Continued Fractions Involving Two Basic Hypergeometric Series

e / aq n  f / aq n

eq   fq  aq 

Fi 

i

i

n

Hi 

q i n , aq i 1 , dq i 1 / b, dq i 1 / c; q  , 4 3  2i 1 2  n 1 2 n  dq , aq / e , aq / f  

i 1 n

n

e / aq n  f / a n

eq   fq  aq  i

i

n

H i 1 

i n

n

e / aq n  f / a n

eq   fq i 1

q i  n , aq i , dq i / b, dq i / c; q  , 4 3  2i 1 n  i / e, aq 1n i / f  dq , aq

 aq 

i 1

n

(9)

i 1 n

(10)

q i  n 1 , aq i 1 , dq i 1 / b, dq i 1 / c; q  , 2i  2 2n i / e, aq 2 n i / f  dq , aq

4 3 

n

(11)

So,

  

H i 1  eq i fqi Fi (1  e / aq)1  f / aq 



1  aq 1  dq / b1  dq 1  dq 1  dq 1  aq i 1

2 i 1

q 2n



q

i 1

i  n 1

 aq

2i  2

i 1

, dq

 q  dq  aq r 0

r 1

r 1 2i  2

i 1

2 n i

r



 

/ c e / aq n  f / aq n aq i  2 2n i / e 1  aq 2n i / f eq i 1

i 1



i 1



 1  dq q    fq  1  dq  n

i  n 1

i 1

n

i 1

2 i 1

n

(12) r

/ b, dq / c r q , / e, aq 2 n i / f r



This gives the following equation:

Fi 1   H i 1 1  e / aq 1  f / aq 1  eq i 1  fq i 

1  aq 1  dq / b1  dq / c 1  dq q 1  dq 1  dq 1  aq / e1  aq / f eq   fq  i 1

2 i 1

i 1

i 1

2i  2

2 n i

i  n 1

2n i

i 1 2 n i

i

n

This can be written as

Fi i  i  , where H i 1 H i 1 Fi 1

n

Fi 1 , H i 1

(13)

are mentions above.

Now using (1) and (2) repeatedly, for i  0 we get (6).

Proof of (b): 1 1  de  2 2  , q  ,  q  , a, b, c, dq n , eq n ;   abc  = 8 7  1  12  2   ,  , q a , q b , q c , e, d 

Where

q  n , a, b, c; q   d , e, f 

4 3 

(15)

.

Let us replace DOI: 10.9790/5728-1303046467

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On Certain Continued Fractions Involving Two Basic Hypergeometric Series 1 1  de  2 2  , q  ,  q  , aq, b, c, dq n 1 , eq n ;   abc  = 8 7  1  12   , 2 , q a , q b , q c , e, dq 

1 1  de  2 2  , q  ,  q  , a, b, c, dq n , eq n ;   abc  8 7  1  12    ,  2 , q a , q b , q c , e, d  1 1   n 1 n de 2 2  , q ,q , aq, b, c, dq , eq ;  abc  8 7  1  12  2  ,   , q a , q b , q c , e, dq  

q  n , aq, b, c; q   dq, e, f 

4 3 

(16)

q  n , a, b, c; q   4 3  d , e, f  n q , aq, b, c; q   4 3  dq, e, f 

=

Now using result (6) on the right hand side we get the result (7).

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[2].

[3].

[4].

[5].

[6].

[7].

Andrews, G.E., Askey R. and Roy, Ranjan. Special Functions, Cambridge University Press, Cambridge, 1999.

Agarwal, R.P. (1996): Resonance of Ramanujan’s Mathematics, vol.II, New Age International (P) Limited, New Delhi.

A.M. Mathai and R.K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and Physical Sciences.

Springer-Verlag, Berlin (1973).

Andrews, G.E. and Berndt, B.C. (2005): Ramanujan’s Lost Notebook,Part1,Springer, New York Gasper, G. and Rahman, M. (1990): Basic hyper geometric series, Cambridge University Press, Cambridge.

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