Some results on vanishing coefficients in infinite product expansions
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Some results on vanishing coefficients in infinite product expansions Nayandeep Deka Baruah1 · Mandeep Kaur1 Received: 10 December 2018 / Accepted: 8 April 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract Recently, Hirschhorn proved that, if ∞
an q n := (− q, − q 4 ; q 5 )∞ (q, q 9 ; q 10 )3∞
n=0
and ∞
bn q n := (− q 2 , − q 3 ; q 5 )∞ (q 3 , q 7 ; q 10 )3∞ ,
n=0
then a5n+2 = a5n+4 = 0 and b5n+1 = b5n+4 = 0. Motivated by the work of Hirschhorn, Tang proved some comparable results including the following: If ∞
cn q n := (− q, − q 4 ; q 5 )3∞ (q 3 , q 7 ; q 10 )∞
n=0
and ∞
dn q n := (− q 2 , − q 3 ; q 5 )3∞ (q, q 9 ; q 10 )∞ ,
n=0
then c5n+3 = c5n+4 = 0 and d5n+3 = d5n+4 = 0.
The first author’s research was partially supported by Grant No. MTR/2018/000157 of Science & Engineering Research Board (SERB), DST, Government of India. Extended author information available on the last page of the article
123
N. D. Baruah, M. Kaur
In this paper, we prove that a5n = b5n+2 , a5n+1 = b5n+3 , a5n+2 = b5n+4 , a5n−1 = b5n+1 , c5n+3 = d5n+3 , c5n+4 = d5n+4 , c5n = d5n , c5n+2 = d5n+2 and c5n+1 > d5n+1 , We also record some other comparable results not listed by Tang. Keywords q-Series expansions · Infinite q-products · Jacobi’s triple product identity · Vanishing coefficients Mathematics Subject Classification Primary 33D15; Secondary 11F33
1 Introduction For complex numbers a and q, with |q| < 1, we define (a; q)∞ :=
∞
(1 − aq k )
k=0
and (a1 , a2 , . . . , an ; q)∞ := (a1 ; q)∞ (a2 ; q)∞ · · · (an ; q)∞ . In this paper, we prove some new results on vanishing coefficients in the series expansions of certain infinite q-products. In the following few paragraphs, we review the work done on this topic by the previous authors. In 1978, Richmond and Szekeres [7] proved that if ∞
αn q n :=
n=0
∞ (q 3 , q 5 ; q 8 )∞ (q, q 7 ; q 8 )∞ and βn q n := 3 5 8 , 7 8 (q, q ; q )∞ (q , q ; q )∞ n=0
then the coefficients α4n+3 and β4n+2 always vanish. They also conjectured that if ∞ n=0
∞ (q 5 , q 7 ; q 12 )∞ (q, q 11 ; q 12 )∞ n γn q := and δ q := , n (q, q 11 ; q 12 )∞ (q 5 , q 7 ; q 12 )∞ n
n=0
then γ6n+5 and δ6n+3 vanish. In [2], Andrews and Bressoud proved the following general theorem, which contains the results of Richmond and Szekeres as special cases.
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Some results on vanishing coefficients in infinite product expansions
Theorem 1.1 (Andrews and Bressoud) If 1 ≤ r < k are relatively prime integers of opposite parity and ∞
(q r , q 2k−r ; q 2k )∞ =: φn q n , (q k−r , q k+r ; q 2k )∞ n=0
then φkn+r (k−r +1)/2 is always zero. In [1], Alladi and Gordon generalized the above theorem as follows: Theorem 1.2 (Alladi and Gordon) Let 1 < m < k and let (s, km) = 1 with 1 ≤ s < mk. Let r ∗ = (k − 1)s and r ≡ r ∗ mod mk with 1 ≤ r < mk. Put r = r ∗ /mk mod k with 1 ≤ r < k. Write ∞
(q r , q mk−r ; q mk )∞ =: μn q n . s mk−s mk (q , q ; q )∞ n=0
Then μn = 0 for n ≡ rr mod k. They also proved the following companion result to Theorem 1.2. Theorem 1.3 (
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