Engel Expansions of q-Series

Engel Expansions of q-Series

Peter Paule

RISC, Linz (Austria)

Algorithms Seminar

October 2, 2000

[summary by Bruno Salvy]

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1 Engel Expansions

A real number A>0 has a unique expansion of the form

A=a₀+

a₁

a₁a₂

a₁a₂a₃

+...,

where the a_i are positive integers with a_i+1³ a_i for i³1. These expansions were called Engel expansions by Perron and their study goes back to Lambert around 1770. Uniqueness of the expansion is not difficult to see, together with the following recurrences from which an iterative algorithm derives:

a_k=ë r_kû+1,

r_k

a_k

a_kr_k+1

, k³1.

The initial conditions are given by a₀<A£ a₀+1 and A-a₀=1/r₁. Rational numbers are characterized by the ultimate stationarity of the sequence (a_i). An obvious example of Engel expansion of a non-rational number is provided by e=exp(1) for which a₀=2 and a_i=i+1 for i>0.

Arnold and John Knopfmacher defined in [4, 5] an analogous notion for Laurent series.

Definition 1 Given a Laurent series A=å_n³nc_nqⁿÎC((q)), and an integer r³0, the q-Engel sequence associated with A and r is the unique sequence (a_i) of polynomials in q^-1 such that

A=a₀+

n³1

q^{-r n}

a₁··· a_n

with the degrees of the a_i obeying deg(a_i+1)³deg(a_i)+r+1.

This definition is motivated by the numerous q-identities involving such expansions. A sample is given in Table 1, using the classical notations

(a;q)₀=1, (a;q)_k=(1-a)(1-aq)···(1-aq^k-1) for k>0

, (a;q)_¥=

k³0

(1-aq^k).

Again, uniqueness is not difficult to check and an iterative algorithm follows from

A_k+1:=q^r(a_kA_k-1), a_k=

é
ê
ê
ë

A_k

ù
ú
ú
û

, k³ 1, (1)

with A₀:=A, a₀=[A] and A₁=q^r(A₀-a₀). The bracket notation corresponds to the integral part of a Laurent series defined by [A]:=å_{n£ n£ 0}c_nqⁿÎC[q^-1].

å

k³0

q

æ
è

k+1

2
ö
ø

(q;q)_k

=

Õ

k>0
(1+q^k),
  (Euler)     (2)

å

k³0

z^kq^k²

(q;q)_k(zq;q)_k

=

1

(z;q)_¥
,
  (Cauchy)     (3)

å

k³0

q^k²

(-q;q²)_k

=

å

k³0

(-1)^k+1q^k

(-q;q)_k
,
  (Fine)     (4)

å

k³0

q^k(3k-1)/2

(q;q)_k(q;q²)_k

=

Õ

k³1

(1-q^10k-6)(1-q^10k-4)(1-q^10k)

1-q^k
,
  (Rogers)     (5)

å

k>0

q^k(3k-1)/2

(q;q)_k-1(q;q²)_k

=

Õ

k³1

(1-q^10k-8)(1-q^10k-2)(1-q^10k)

1-q^k
,
  (Rogers)     (6)

å

k³0

q^k(2k-1)

(q;q)_2k

=

Õ

k>0
(1+q^k),
  (Slater)     (7)

å

k³0

q^k²

(q;q)_k

=

1

(q;q⁵)_¥(q⁴;q⁵)_¥
,
  (1st Rogers--Ramanujan)     (8)

å

k³0

q^k²+k

(q;q)_k

=

1

(q²;q⁵)_¥(q³;q⁵)_¥
,
  (2nd Rogers--Ramanujan)     (9)

å

k³0

q^2k²

(q;q)_2k

=

Õ

k>0,   kº±2, ±3, ±4, ±5(mod 16 )

1

1-q^k
,
   (Slater)     (10)

å

k³0

q^2k²+2k

(q;q)_2k+1

=

Õ

k>0   kº±1, ±4, ±6, ±7(mod 16 )

1

1-q^k
,
   (Slater)     (11)

Table 1: q-identities involving q-Engel expansions.

2 Engel Guessing

Equipped with (1), it is very natural to implement a package computing q-Engel sequences of Laurent series. Such a package opens the way to experimental mathematics with q-Engel expansions [2]. For instance, starting from a truncation of the series expansion of the right-hand side of (2) (a special case of an identity due to Euler) and using r=0, the package outputs

(q;q)₁

q³

(q;q)₂

q⁶

(q;q)₃

+O(q¹⁰),

from which the left-hand side is easily guessed. The task of proving such an identity still requires human work.

Using r=1 on the same series does not reveal any pattern. However, with r=2, one gets

(q;q)₂

q⁶

(q;q)₄

q¹⁵

(q;q)₆

+O(q²⁸),

from which it is easy to conjecture the general formula (7).

3 Identities of Rogers--Ramanujan Type

In one of his independent proofs of the Rogers--Ramanujan identities (8--9), Schur introduced two sequences of polynomials

d_m=

(-1)^kq^k(5k-3)/2

é
ê
ê
ê
ë

m-1

m+1-5k

ù
ú
ú
ú
û

, e_m=

(-1)^kq^k(5k+1)/2

é
ê
ê
ê
ë

m-1

m-1-5k

ù
ú
ú
ú
û

, m³1,

with e₀=0 and d₀=1 in terms of the Gaussian polynomials

é
ë

ù
û

ì
ï
í
ï
î

(q;q)_n

(q;q)_k(q;q)_n-k

if 0£ k£ n,

otherwise.

The sequences d_m and e_m appear in the recent generalization of the Rogers--Ramanujan identities due to Garrett, Ismail and Stanton [3]:

n=0

q^n²+mn

(q;q)_n

(-1)^mq

æ
è

ö
ø

d_m

(q;q⁵)_¥(q⁴;q⁵)_¥

(-1)^mq

æ
è

ö
ø

e_m

(q²;q⁵)_¥(q³;q⁵)_¥

. (12)

Setting m=0, m=1 in this formula yields (8) and (9).

The left-hand side of (12) is the q-Engel expansion of the right-hand side for r=0, which motivates [1] in looking for a q-Engel ``proof'' of this identity. For this, it is sufficient to prove that the sequence a_n=q^-(2n+m-1)-q^-(n+m-1) is the corresponding q-Engel sequence. Defining

A₀=A, A_n=(-1)^mq

æ
è

ö
ø

-(m-1)(n-1)

j>m

q^jn(d_me_j-d_je_m) for n³1,

the proof consists in showing that a_nA_n=1+A_n+1 and a_n=[1/A_n], together with correct initial conditions. In view of (14) below, this is not too difficult, but technical (see [1] for details).

Schur proved that both d_m and e_m satisfy the recurrence

c_m+2=c_m+1+q^mc_m, m³0. (13)

Nowadays, this identity is proved automatically by invoking the q-WZ algorithm [7] and this leads to the first purely automatic elementary proof of the Rogers--Ramanujan identity [6]. In view of this recurrence, d_m and e_m are nothing but q-analogues of the Fibonacci numbers. It turns out that a generalization of the Cassini identity, namely

F_m-1F_m+k-F_m+k-1F_m=(-1)^mF_k,

admits a q-analogue in terms of e_m and d_m:

d_me_m+k-d_m+ke_m=(-1)^mq

æ
è

ö
ø

j³0

é
ë

k-1-j

ù
û

q^j²+mj. (14)

This identity can be proved automatically from (13) by univariate D-finite closure properties (m being fixed). In fact, a non-Engel proof of (12) follows from letting k tend to infinity in (14) in view of Schur's limit formulae

d_¥

(q;q)_¥

(-1)^kq^k(5k-3)/2

(q²;q⁵)_¥(q³;q⁵)_¥

e_¥

(q;q)_¥

(-1)^kq^k(5k+1)/2

(q;q⁵)_¥(q⁴;q⁵)_¥

The infinite products are obtained by Jacobi's triple product identity, which also admits a simple computer proof [6].

4 A New Identity Discovered by Engel Guessing

The identities (10) and (11) can be conjectured by Engel guessing after first multiplying the product by 1-q. An Engel proof is also available [2] using the Santos polynomials defined by

S_m=

q^4j²-j

é
ê
ê
ê
ë

m+1-4j

ù
ú
ú
ú
û

, T_m=

q^4j²-3j

é
ê
ê
ê
ë

m+2-4j

ù
ú
ú
ú
û

whose limits S_¥ and T_¥ when m®¥ are precisely the right-hand sides of (10) and (11).

In view of (12), a natural idea consists in experimenting with q-Engel expansions of S_nT_¥-T_nS_¥ or variations of it. It turns out that a pattern readily emerges leading to conjecturing the following generalization of (10) and (11):

S_nT_¥-T_nS_¥=qⁿ(q;q²)_n

k³0

q^2k²+2(n+1)k

(q;q)_2k+1

Again, a possible proof [2] consists in relying on a finite version, namely

S_nT_n+m-T_nS_n+m=qⁿ(q;q²)_n

k³0

é
ë

2k+1

ù
û

q^2k²+2(n+1)k

5 Conclusion

Engel expansions are a new way of looking at q-identities which allows for easy computer experiments and hence should lead to many discoveries. A pending issue is to make q-Engel proving into an algorithmic task.

References

[1]: Andrews (George E.), Knopfmacher (Arnold), and Paule (Peter). -- An infinite family of Engel expansions of Rogers-Ramanujan type. Advances in Applied Mathematics, vol. 25, n°1, 2000, pp. 2--11.
[2]: Andrews (George E.), Knopfmacher (Arnold), Paule (Peter), and Zimmermann (Burkhard). -- Engel expansions of q-series by computer algebra. In Alladi (K.) (editor), q-Series. Kluwer Series Developments in Mathematics. -- Kluwer, 2000. Proceedings of the Conference ``q-Series'', Gainesville, November 1999.
[3]: Garrett (Kristina), Ismail (Mourad E. H.), and Stanton (Dennis). -- Variants of the Rogers-Ramanujan identities. Advances in Applied Mathematics, vol. 23, n°3, 1999, pp. 274--299.
[4]: Knopfmacher (Arnold) and Knopfmacher (John). -- Inverse polynomial expansions of Laurent series. Constructive Approximation, vol. 4, n°4, 1988, pp. 379--389.
[5]: Knopfmacher (Arnold) and Knopfmacher (John). -- Inverse polynomial expansions of Laurent series. II. In Proceedings of the 3rd International Congress on Computational and Applied Mathematics (Leuven, 1988), vol. 28, pp. 249--257. -- 1989.
[6]: Paule (Peter). -- Short and easy computer proofs of the Rogers-Ramanujan identities and of identities of similar type. Electronic Journal of Combinatorics, vol. 1, 1994. -- Research Paper 10. 9 pages.
[7]: Wilf (Herbert S.) and Zeilberger (Doron). -- An algorithmic proof theory for hypergeometric (ordinary and ``q'') multisum/integral identities. Inventiones Mathematicae, vol. 108, n°3, 1992, pp. 575--633.

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