- Email: [email protected]
A Generalization of Two q-Identities of Andrews
Journal of Combinatorial Theory, Series A 95, 381386 (2001) doi:10.1006jcta.2001.3180, available online at http:www.idealibrary.com on
NOTE A Generalization of Two q-Identities of Andrews Kuo-Jye Chen Department of Mathematics, National Changhua University of Education, Changhua 50058, Taiwan E-mail: kjcmath.math.ncue.edu.tw
and H. M. Srivastava Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada E-mail: harimsrimath.uvic.ca Communicated by George Andrews Received June 1, 2000; published online June 18, 2001
The main object of the present paper is to give a unification (and generalization) of two interesting q-identities which were proven recently by George E. Andrews. Some related results involving the Fibonacci numbers are also considered. 2001 Academic Press
Key Words: q-identities; RogersRamanujan identities; Schur's identity; Fibonacci numbers; Gaussian polynomials; rational functions.
1. INTRODUCTION AND PRELIMINARIES From a truly amazing heuristic investigation of the behavior of the familiar RogersRamanujan identities near the unit circle, Daan Krammer stated a conjecture involving the sum: n
K n :=1+2 : (&1) k q k=1
( ) 2k&1 , k
& k 2
_
&
381 0097-316501 35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved.
382
NOTE
where the Gaussian polynomial is given by
n
_k& =
{
0,
if k<0 or k>n
1,
if k=0 or k=n
(q; q) n , (q; q) n&k (q; q) k
if 0
with, as usual, (*; q) 0 =1
(*; q) n =(1&*)(1&*q) } } } (1&*q n&1 )(n # N),
and
N being the set of natural numbers. John Greene [3] gave a proof of Krammer's conjecture by making use of Schur's identity:
D n+1 :=
:
qj
2
0 2 j n
n& j = : (&1) * q (12) *(5*+1) j *=&
_ &
n n&5* 2
__ &&
,
where, in addition, D 0 =0, [x] being the greatest integer function. Andrews [1] found the following interesting identities connecting K n and D n : n
K n :=1+2 : (&1) k q
( ) 2k&1 k
& k 2
k=1 n
=&1+2 : (&1) n&k q
_
& n&k 2
(
k=0 n
& n&k =&1+2 : (&1) n&k q ( 2 k=0
& ) 2n+1 _ n&k & D ) 2n D _n&k&
2k+2
2k+1
.
(1)
As q Ä 1, these identities lead to the following identities for Fibonacci numbers [1, p. 73]: n
: (&1) k k=0
\
n 2k&1 2n+1 = : (&1) n&k F 2k+2 k n&k k=0
+
n
= : (&1) n&k k=0
\ + 2n \n&k+ F
2k+1
,
where F 1 =F 2 =1
and
F n =F n&1 +F n&2 for
n # N"[1, 2].
(2)
383
NOTE
By means of certain well-known generating functions for the Fibonacci numbers (cf., e.g., [2, p. 336]), the three sums in (2) are readily seen to be the coefficients of x n in the following three rational functions, respectively: (1&x) 2n+1 1 1&x =(1&x) 2n . =(1&x) 2n+1 2 2 (1&x) &x 1&3x+x 1&3x+x 2 This evidently completes an alternative (direct) proof of the identities (2). Next, by considering a more general rational function, (1&x) m , 1&3x+x 2 where m is an arbitrary complex number, we can obtain the following generalization of the known identities (2): n
: (&1) k k=0
\
n m&2(n&k)&2 m = : (&1) n&k F 2k+2 k n&k k=0
+
n
= : (&1) n&k k=0
\ + m&1 \ n&k + F
2k+1
.
(3)
When m=2n+1, (3) readily yields (2).
2. THE MAIN RESULT AND ITS CONSEQUENCES Formula (3) and Andrews's proof [1] of Formula (1) lead us eventually to the following unification (and generalization) of (1) and (3): Theorem. Let m and r be arbitrary complex numbers such that m+r n whenever m+r is an integer. Then n m&2(n&k)&2 & k : (&1) k q ( 2 ) k k=0
_
n
= : (&1) n&k q k=0
& n&k 2
(
&
) m+r A 2k+2+r , n&k
_ &
(4)
where k
A 2k+2+r := : q j j=0
2 + (m&2n&1) j
_
2k+2+r& j&1 j
&
(n # N 0 :=N _ [0]). (5)
384
NOTE
Remark 1. The first equality of (1) can be deduced as a special case of (4) when m=2n+1 (n # N)
and
r=0.
Remark 2. In its special case when m=2n+1 (n # N)
and
r=&1,
our assertion (4) yields the second equality of (1). Remark 3. When q Ä 1, the special cases r=0 and r=&1 of our assertion (4) reduce, respectively, to the first and second equalities of (3). Proof of the Theorem.
We first recall that
(1&q : )(1&q :&1 ) } } } (1&q :&k+1 ) (q :&k+1; q) k : = = (1&q k )(1&q k&1 ) } } } (1&q) (q; q) k k
_&
(k # N 0 ).
(6)
Now, by applying each of the formulas:
_
m&2k&2 (q m&n&k&1; q) n&k = n&k (q; q) n&k
&
=
(q m&n+k+r+1; q) n&k (q m&n&k&1; q) n&k , (q; q) n&k (q m&n+k+r+1; q) n&k
(7)
N j (a; q) N N (ba; q) j =: (&a) j q( 2 ) (b; q) N j=0 j (b; q) j
_&
(see [4, p. 247, Eq. (IV.3)]),
(8)
and &
\
n&k+ j j n&k +(n&k+ j ) j+ =& + j 2, 2 2 2
+
\+ \ +
we observe that the first member of the q-identity (4) is n
: (&1) k q k=0
( ) m&2(n&k)&2 k
& k 2
_
n
= : (&1) n&k q k=0 n
& n&k 2
(
) m&2k&2 n&k
_
& n&k (q = : (&1) n&k q ( 2 ) k=0
& &
m&n+k+r+1
; q) n&k (q m&n&k&1; q) n&k (q; q) n&k (q m&n+k+r+1; q) n&k
(9)
385
NOTE n
= : (&1) n&k q
& n&k 2
(
k=0 n&k
_: j=0 n
m&n+k+r+1 ; q) n&k ) (q (q; q) n&k
n&k (q 2k+2+r; q) j (m&n&k&1) j+ ( j ) 2 (&1) j q m&n+k+r+1 (q ; q) j j
_ &
n&k
=:
& n&k +(m&n&k&1) j+ ( j ) 2 : (&1) n&k+ j q ( 2 )
k=0 j=0
_
(q m&n+k+r+1+ j; q) n&k& j (q 2k+2+r; q) j (q; q) n&k& j (q; q) j
n
n& j
=:
: (&1) n&k+ j q
& n&k +(m&n&k&1) j+ j 2 2
(
)
( )
j=0 k=0
_
m+r
_n&k& j&_
n
2k+2+r+ j&1 j
&
n
=:
& n&k+ j +(m&n&k+ j&1) j+ ( j ) 2 : (&1) n&k q ( 2 )
j=0 k= j
_
m+r n&k
_ &_
n
2k+2+r&j&1 j
k
=:
: (&1) n&k q
&
& n&k+j +(n&k+j ) j+ j +(m&2n&1) j 2 2
(
)
( )
k=0 j=0
_
m+r
_ n&k &_
n
=: k=0
2k+2+r&j&1 j
k m+r & n&k +j 2 +(m&2n&1) j : (&1) n&k q ( 2 ) n&k j=0
n
= : (&1) n&k q
& n&k 2
(
k=0 n
= : (&1) n&k q k=0
&
& n&k 2
(
) m+r n&k
k
_ & ) m+r _ n&k & A
: qj
2k+2+r&j&1
_ &_ j & 2k+2+r&j&1 _ j &
2 +(m&2n&1) j
j=0
2k+2+r
,
which evidently completes the proof of the theorem.
ACKNOWLEDGMENTS The present investigation was completed during the second-named author's visit to the National Changhua University of Education at Changhua in March 2000. This work was supported, in part, by the National Science Council of the Republic of China under Grant NSC 87-2115-M-018-004 and, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
386
NOTE
REFERENCES 1. G. E. Andrews, On the GreeneKrammer theorem and related identities, Ganita 43 (1992), 6973. 2. R. L. Graham, D. E. Knuth, and O. Patashnik, ``Concrete Mathematics: A Foundation for Computer Mathematics,'' AddisonWesley, Reading, MA, 1994. 3. J. Greene, On a conjecture of Krammer, J. Combin. Theory Ser. A 56 (1991), 309311. 4. L. J. Slater, ``Generalized Hypergeometric Functions,'' Cambridge Univ. Press, Cambridge LondonNew York, 1966.
NOTE A Generalization of Two q-Identities of Andrews Kuo-Jye Chen Department of Mathematics, National Changhua University of Education, Changhua 50058, Taiwan E-mail: kjcmath.math.ncue.edu.tw
and H. M. Srivastava Department of Mathematics and Statistics, University of Victoria, Victoria, British Columbia V8W 3P4, Canada E-mail: harimsrimath.uvic.ca Communicated by George Andrews Received June 1, 2000; published online June 18, 2001
The main object of the present paper is to give a unification (and generalization) of two interesting q-identities which were proven recently by George E. Andrews. Some related results involving the Fibonacci numbers are also considered. 2001 Academic Press
Key Words: q-identities; RogersRamanujan identities; Schur's identity; Fibonacci numbers; Gaussian polynomials; rational functions.
1. INTRODUCTION AND PRELIMINARIES From a truly amazing heuristic investigation of the behavior of the familiar RogersRamanujan identities near the unit circle, Daan Krammer stated a conjecture involving the sum: n
K n :=1+2 : (&1) k q k=1
( ) 2k&1 , k
& k 2
_
&
381 0097-316501 35.00 Copyright 2001 by Academic Press All rights of reproduction in any form reserved.
382
NOTE
where the Gaussian polynomial is given by
n
_k& =
{
0,
if k<0 or k>n
1,
if k=0 or k=n
(q; q) n , (q; q) n&k (q; q) k
if 0
with, as usual, (*; q) 0 =1
(*; q) n =(1&*)(1&*q) } } } (1&*q n&1 )(n # N),
and
N being the set of natural numbers. John Greene [3] gave a proof of Krammer's conjecture by making use of Schur's identity:
D n+1 :=
:
qj
2
0 2 j n
n& j = : (&1) * q (12) *(5*+1) j *=&
_ &
n n&5* 2
__ &&
,
where, in addition, D 0 =0, [x] being the greatest integer function. Andrews [1] found the following interesting identities connecting K n and D n : n
K n :=1+2 : (&1) k q
( ) 2k&1 k
& k 2
k=1 n
=&1+2 : (&1) n&k q
_
& n&k 2
(
k=0 n
& n&k =&1+2 : (&1) n&k q ( 2 k=0
& ) 2n+1 _ n&k & D ) 2n D _n&k&
2k+2
2k+1
.
(1)
As q Ä 1, these identities lead to the following identities for Fibonacci numbers [1, p. 73]: n
: (&1) k k=0
\
n 2k&1 2n+1 = : (&1) n&k F 2k+2 k n&k k=0
+
n
= : (&1) n&k k=0
\ + 2n \n&k+ F
2k+1
,
where F 1 =F 2 =1
and
F n =F n&1 +F n&2 for
n # N"[1, 2].
(2)
383
NOTE
By means of certain well-known generating functions for the Fibonacci numbers (cf., e.g., [2, p. 336]), the three sums in (2) are readily seen to be the coefficients of x n in the following three rational functions, respectively: (1&x) 2n+1 1 1&x =(1&x) 2n . =(1&x) 2n+1 2 2 (1&x) &x 1&3x+x 1&3x+x 2 This evidently completes an alternative (direct) proof of the identities (2). Next, by considering a more general rational function, (1&x) m , 1&3x+x 2 where m is an arbitrary complex number, we can obtain the following generalization of the known identities (2): n
: (&1) k k=0
\
n m&2(n&k)&2 m = : (&1) n&k F 2k+2 k n&k k=0
+
n
= : (&1) n&k k=0
\ + m&1 \ n&k + F
2k+1
.
(3)
When m=2n+1, (3) readily yields (2).
2. THE MAIN RESULT AND ITS CONSEQUENCES Formula (3) and Andrews's proof [1] of Formula (1) lead us eventually to the following unification (and generalization) of (1) and (3): Theorem. Let m and r be arbitrary complex numbers such that m+r n whenever m+r is an integer. Then n m&2(n&k)&2 & k : (&1) k q ( 2 ) k k=0
_
n
= : (&1) n&k q k=0
& n&k 2
(
&
) m+r A 2k+2+r , n&k
_ &
(4)
where k
A 2k+2+r := : q j j=0
2 + (m&2n&1) j
_
2k+2+r& j&1 j
&
(n # N 0 :=N _ [0]). (5)
384
NOTE
Remark 1. The first equality of (1) can be deduced as a special case of (4) when m=2n+1 (n # N)
and
r=0.
Remark 2. In its special case when m=2n+1 (n # N)
and
r=&1,
our assertion (4) yields the second equality of (1). Remark 3. When q Ä 1, the special cases r=0 and r=&1 of our assertion (4) reduce, respectively, to the first and second equalities of (3). Proof of the Theorem.
We first recall that
(1&q : )(1&q :&1 ) } } } (1&q :&k+1 ) (q :&k+1; q) k : = = (1&q k )(1&q k&1 ) } } } (1&q) (q; q) k k
_&
(k # N 0 ).
(6)
Now, by applying each of the formulas:
_
m&2k&2 (q m&n&k&1; q) n&k = n&k (q; q) n&k
&
=
(q m&n+k+r+1; q) n&k (q m&n&k&1; q) n&k , (q; q) n&k (q m&n+k+r+1; q) n&k
(7)
N j (a; q) N N (ba; q) j =: (&a) j q( 2 ) (b; q) N j=0 j (b; q) j
_&
(see [4, p. 247, Eq. (IV.3)]),
(8)
and &
\
n&k+ j j n&k +(n&k+ j ) j+ =& + j 2, 2 2 2
+
\+ \ +
we observe that the first member of the q-identity (4) is n
: (&1) k q k=0
( ) m&2(n&k)&2 k
& k 2
_
n
= : (&1) n&k q k=0 n
& n&k 2
(
) m&2k&2 n&k
_
& n&k (q = : (&1) n&k q ( 2 ) k=0
& &
m&n+k+r+1
; q) n&k (q m&n&k&1; q) n&k (q; q) n&k (q m&n+k+r+1; q) n&k
(9)
385
NOTE n
= : (&1) n&k q
& n&k 2
(
k=0 n&k
_: j=0 n
m&n+k+r+1 ; q) n&k ) (q (q; q) n&k
n&k (q 2k+2+r; q) j (m&n&k&1) j+ ( j ) 2 (&1) j q m&n+k+r+1 (q ; q) j j
_ &
n&k
=:
& n&k +(m&n&k&1) j+ ( j ) 2 : (&1) n&k+ j q ( 2 )
k=0 j=0
_
(q m&n+k+r+1+ j; q) n&k& j (q 2k+2+r; q) j (q; q) n&k& j (q; q) j
n
n& j
=:
: (&1) n&k+ j q
& n&k +(m&n&k&1) j+ j 2 2
(
)
( )
j=0 k=0
_
m+r
_n&k& j&_
n
2k+2+r+ j&1 j
&
n
=:
& n&k+ j +(m&n&k+ j&1) j+ ( j ) 2 : (&1) n&k q ( 2 )
j=0 k= j
_
m+r n&k
_ &_
n
2k+2+r&j&1 j
k
=:
: (&1) n&k q
&
& n&k+j +(n&k+j ) j+ j +(m&2n&1) j 2 2
(
)
( )
k=0 j=0
_
m+r
_ n&k &_
n
=: k=0
2k+2+r&j&1 j
k m+r & n&k +j 2 +(m&2n&1) j : (&1) n&k q ( 2 ) n&k j=0
n
= : (&1) n&k q
& n&k 2
(
k=0 n
= : (&1) n&k q k=0
&
& n&k 2
(
) m+r n&k
k
_ & ) m+r _ n&k & A
: qj
2k+2+r&j&1
_ &_ j & 2k+2+r&j&1 _ j &
2 +(m&2n&1) j
j=0
2k+2+r
,
which evidently completes the proof of the theorem.
ACKNOWLEDGMENTS The present investigation was completed during the second-named author's visit to the National Changhua University of Education at Changhua in March 2000. This work was supported, in part, by the National Science Council of the Republic of China under Grant NSC 87-2115-M-018-004 and, in part, by the Natural Sciences and Engineering Research Council of Canada under Grant OGP0007353.
386
NOTE
REFERENCES 1. G. E. Andrews, On the GreeneKrammer theorem and related identities, Ganita 43 (1992), 6973. 2. R. L. Graham, D. E. Knuth, and O. Patashnik, ``Concrete Mathematics: A Foundation for Computer Mathematics,'' AddisonWesley, Reading, MA, 1994. 3. J. Greene, On a conjecture of Krammer, J. Combin. Theory Ser. A 56 (1991), 309311. 4. L. J. Slater, ``Generalized Hypergeometric Functions,'' Cambridge Univ. Press, Cambridge LondonNew York, 1966.