Internat. J. Math. & Math. Sci.
VOL. 13 NO. 4 (1990) 741-746
741
ON GENERALIZATION OF CONTINUED FRACTION OF GAUSS
REMY Y. DENIS
Department of Mathematics
University of Gorakhpur
Gorakhpur- 273009
INDIA
(Received May 10, 1988)
ABSTRACT.
In this paper we establish a continued fraction represetatlon for the ratio
qf two basic bilateral hypergeometrlc series
fraction for the ratio of t:
2FI
22
s
which generalize
Gauss’ continued
’s.
KEY WORDS AND PHRASES. Continued fractions and hypergeometric series.
1980 AMS SUBJECT CLASSIFICATION CODE.
IIA55.
INTRODUCTION.
Gauss (see Wall [3] and also Jones and Thron [2], gave the following continued
fraction involving the ratio of two Gausslan
2Fl’s,
F[a’b+l;Zc+l ]/F[a’b;Z]c
1-
(1.1)
a(c-b)z/c(c+1)
(b+l) (c-a+l)z/(c+l) (c+2)
(a+ I) (c-b+ 1) z{(c+2) (c+3)1(b+2) (c-a+2)z/(c+3) (c+4)
I-
1-
where
2F
in which the symbol
[a,;
T
[a]n
=o
[l]n[Y]n
stands for a(a+1)(a+2)...(a+n-1) and [a]
In this paper we establlsh the continued fraction for the ratio
22
,Y
22
,Tq
where
2)2
,
[a] [Sl x
[]n [’f]n
n
I.
REMY Y. DENIS
742
.
where
[=]
n
_-- [a;q]
n
(1-a)
-aq ).
(l -aq
n-I
),[a] o
The other notations appearing in this paper carry their usual meaning.
2.
MAIN RESULT.
In this paper we establish the following result
xB
Ao Co
where for i
B
xD
/
AI
/
/
xD
CI
xB
+
/
2
+
C2
O, I, 2, 3,
A
B
C
i
(l_Sql) (_y21+l
6)
(l_Tq2i)(q i+l
6)
qi+l_ :(l_.qi) (l__ql) (_yql)
-q 2i+1) (I -Yq 2i) q i+l
i
,(I-Ri) (yq2i+2
I
(l-q2i+l)(aq
6
6)
i+l
6)
and
i+
D
i
PROOF of (2.1).
I(I
_qi+ l)(l-aqi)(a-yq
q
2i+1 (l_Tq2t+2
(i-Vq
i+l
_)
(1t+1
6)
It is easy to see that the following relation is true (for non-
negative integral t),
i,i;x
22
q 2i
6,
Ai 22
,
J
+ xB
Tq
2i+1
6,
6
Tq
2i+2
q
and T by Yq in it,
i+l
i
2r2
(2.2)
2P2
in, (2.2) avd then replacing B by
No, interchanging a and
we get
i
Tq
21+1
+
Ct 22
6
Tq
2t+2
xDt
(2.3)
2q2
yq
21+
ON GENERALIZATION OF CONTINUED FRACTION OF GAUSS
Now from (2.2) for i
22
743
o, we get
22
6 ,-f
q
+
A
22
,
6,Yq
Tq
xB
+
A
xD
from (2.3) with
+
A
xD
+
A
, yq4
2J2
Tq3
6
from (2.2) with i
xB
o + C
o
o
xD
xB
o
+ A + C
xD
xB
2
+ A2 + C 2
(by repeated application of (2.2) and (2,3)).
3.
Tfs proves (2.1).
SPECIAL CASES.
Here we shall reduce certain interesting special cases of (2.1).If in (2.1) we
take 6
q, we get
2
Yq
xv o
xl
+
1+
where for i
xv
xrl
+
+
+
x2
+
(3.1)
+
O, I, 2,
Pt
q
t
(1-Cl t) (Yq t-I) / (1-’Yq 2t) (1 -’Yq 21+1
and
v
If q
in
t
q
t
1-6q
i+l
Yq
t+1
-) /
-q 2t+1)
-yq 21+2 ).
(3.1), we get (I.I), the continued fraction of Gauss.
If in (3.1) we take 6-
and replace Y by Y/q, we get,
744
REMY Y. DENIS
vo
Bo
xv
xB
+
+
(3.2)
+
+
+
+
O, l, 2,
where for
B
i) (l-Yq i-I )/(1-Yq 2i-1 )(1-yq 2i
i
q (1-cq
i
and
v
=q
i
i
(1-q
i+l
I,
Now, if in (3.2) we let q
I; x
i
2i+1
)(1-yq /oO/(1-yq 2i) (1-yq
we get the following known result
[2]
1
xo
x
no
x2
xn
(3.3)
-"
O, I, 2,
where for I
i
(’+i) (Y+i- 1) / (+2i- 1) (y+2i)
"i
(i+ 1) (-c+i) / (+2i) (+2i+ 1)
and
and replace x by xq/e and then let s
o in (3.2)
If we put Y
(R), we get the
following interesting result
v
(-)
L
n
q
n(n+l)/2
x
n
n,o
x_x_q_
+
+
z_R_
+
+
xq3(q3-1)
5
xq2(q2-1)_
3
_xq(q_ 1)
x_x_q_
+
(3.4)
+
+
(3.2) we get a continued fraction representation
q in
x] which, when q
I, yields the continued fraction representation for
general binomial (l-x)
If
for
i#o
we
take Y
is;
q, y
Again, if we take a
fraction
representation
for
q
2#i
2
and replace x by -x in
2
[q,q;q ;-x] which,
when q
(3.2), we get a continued
yields
fraction representation for
xl
log (l+x)
F
[I,12; -x]
Similarly, we can get the continued fraction representation for
log
(l+x)
2x F
I1/2, ;x]
3/2
the
continued
ON GENERALIZATION OF CONTINUED FRACTION OF GAUSS
Further,
we
if
take
(3.1),
o in
get
we
the
following
745
result
after some
s impliflcatlon,
where for i
O, I, 2
i
i
q (q -)/(1-q
v
q 2i+I 1-Sq i+l
2i) (1-q 2i+1
and
1-q
2i+l)
-q 2i+2
The above (3.5) is the q-analogue of a known result [2].
Again, setting S
q;x]
expression
in
(3.5) we get the continued fraction representation for
from which one can,
for
q-exponential
for
=I, deduce
function
eq(x)
the corrsponding continued
which
raction representation for exponential function e
z
in
when q
yields
turn
the
fraction
continued
[2].
A number of other interesting special cases could also be deduced,
The reader is
referred to Wall [I] and Jones [2].
ACKNOWLEDGEMENT.
This work has been done under a Research Scheme awarded by
C.S.I.R., Government
of India, New Delhi for which the author expresses his gratitude.
I
am
thankful
to
Professor
R.P.
Agarwal
for
his
encouragement
during
the
preparation of this paper.
REFERENCES
I.
WALL, H.S., Anal[tic theory
York (1948).
2.
JONES, W.B. and THROW, W.J., Continued Fractions, Analytic theory and
appl cl atlons
3.
o.f
continued fractions, D. Van Nostrand Co. Inc. New
Enc clo edla of Mathematics and its application,
I__I
Addison-
Wesley Publishing Company (1980).
ANDREWS, G.E., Ramaujan’s "LOST" Notebook Ill. The Rogers-Ramanujan Continued
Fraction. Advances in Mathematics,
41(2), (1981), 186-208.