A q–ANALOGUE OF A FORMULA OF HERNANDEZ OBTAINED BY INVERTING A RESULT OF DILCHER

HELMUT PRODINGER

Abstract. We prove a q–analogue of the formula

∑_{1≤k≤n} \binom{n}{k}(−1)^{k−1} ∑_{1≤i_{1}≤i_{2}≤···≤i_{m}=k} \frac{1}{i_{1}i_{2} … i_{m}} = ∑_{1≤k≤n} \frac{1}{k^{m}}

by inverting a formula due to Dilcher.

1. The identities

Hernández in [6] proved the following identity:

∑_{1≤k≤n} \binom{n}{k}(−1)^{k−1} ∑_{1≤i_{1}≤i_{2}≤···≤i_{m}=k} \frac{1}{i_{1}i_{2} … i_{m}} = ∑_{1≤k≤n} \frac{1}{k^{m}} .  (1)

However this identity does not really require a proof, since we will show that it is just an inverted form of an identity of Dilcher [3];

∑_{1≤k≤n} \binom{n}{k}(−1)^{k−1} \frac{1}{k^{m}} = ∑_{1≤i_{1}≤i_{2}≤···≤i_{m}≤n} \frac{1}{i_{1}i_{2} … i_{m}} .  (2)

Define for k ≥ 1

a_{k} := − ∑_{1≤i_{1}≤i_{2}≤···≤i_{m}=k} \frac{1}{i_{1}i_{2} … i_{m}}  and b_{k} := \frac{1}{k^{m}} ,

and a_{0} = b_{0} = 0, then the identities are

\begin{aligned}∑_{0≤k≤n} \binom{n}{k}(−1)^{k}a_{k} & = ∑_{0≤k≤n} b_{k}, \\ ∑_{0≤k≤n} \binom{n}{k}(−1)^{k}b_{k} & = ∑_{0≤k≤n} a_{k}. \end{aligned}  (3)

They are inverse relations, as can be seen by introducing ordinary generating functions A(z) = ∑ a_{n}z^{n} and B(z) = ∑ b_{n}z^{n}. Then they are

Date: October 11, 1999.

This research was conducted while the author was a host of the projet Algo at INRIA, Rocquencourt. The funding came from the Austrian–French “Amadée” cooperation.

\frac{1}{1 − z} A(\frac{z}{z − 1}) = \frac{1}{1 − z} B(z),
\frac{1}{1 − z} B(\frac{z}{z − 1}) = \frac{1}{1 − z} A(z).

However

w = \frac{z}{z − 1} ⟷ z = \frac{w}{w − 1},

and the proof is finished. An alternative argument that will be useful in the sequel when we do the q–analogue, is as follows. We assume a_{0} = b_{0} = 0 and take differences in (3) of the lines indexed with n resp. n − 1; then we have to prove that

b_{n} = ∑_{1≤k≤n} \binom{n − 1}{k − 1}(−1)^{k}a_{k} ⟺ a_{n} = ∑_{1≤k≤n} \binom{n − 1}{k − 1}(−1)^{k}b_{k}.

Now in this form this is a traditional inverse relation; see e. g. [7]. An explicit argument will follow in the next section for the q–instance.

We note that Dilcher’s sum appears also in disguised form in [4].

2. A q–analogue

Dilcher’s formula (2) was only a corollary of his elegant q–version;

∑_{1≤k≤n} \left[{n \atop k}\right]_{q}(−1)^{k−1} \frac{q^{\binom{k+1}{2}+(m−1)k}}{(1 − q^{k})^{m}} = ∑_{1≤i_{1}≤i_{2}≤···≤i_{m}≤n} \frac{q^{i_{1}}}{1 − q^{i_{1}}} … \frac{q^{i_{m}}}{1 − q^{i_{m}}} .

Here, \left[{n \atop k}\right]_{q} denotes the Gaussian polynomial

\left[{n \atop k}\right]_{q} = \frac{(q; q)_{n}}{(q; q)_{k}(q; q)_{n−k}}

with

(x; q)_{n} := (1 − x)(1 − xq) … (1 − xq^{n−1}).

Apart from Dilcher’s paper [3], the article [2] is also of some relevance in this context.

Therefore it is a natural question to find a q–analogue of Hernández’ formula, or, what amount to the same, to find the appropriate inverse relations for the q–analogues.

We state them in the following lemma.

Lemma 1. Assume that a_{0} = b_{0} = 0, then

\begin{aligned}∑_{1≤k≤n} b_{k} & = ∑_{1≤k≤n} \left[{n \atop k}\right]_{q}(−1)^{k}q^{\binom{k}{2}}a_{k}, \\ ∑_{1≤k≤n} q^{−k}a_{k} & = ∑_{1≤k≤n} \left[{n \atop k}\right]_{q}(−1)^{k}q^{−kn+\binom{k}{2}}b_{k}. \end{aligned}  (4)

A q–ANALOGUE OF A FORMULA OF HERNANDEZ 3

Proof. First note that it is much easier to prove this than to find it (it took me a few hours, but I am not too experienced).

After a first version of this note was circulated, O. Warnaar kindly informed me that this lemma would follow from results in [5]. Nevertheless, it is perhaps instructive to provide a direct and independent proof.

Taking again differences in (4), we have to prove that

b_{n} = ∑_{1≤k≤n} \left[{n − 1 \atop k − 1}\right]_{q}(−1)^{k}q^{\binom{k}{2}}a_{k} ⟺ a_{n} = ∑_{1≤k≤n} \left[{n − 1 \atop k − 1}\right]_{q}(−1)^{k}q^{−(n−1)k+\binom{k}{2}}b_{k}.(5)

Now

∑_{1≤k≤n}\left[{n − 1 \atop k − 1}\right]_{q}(−1)^{k}q^{−(n−1)k+\binom{k}{2}}b_{k}

= ∑_{1≤k≤n} \left[{n − 1 \atop k − 1}\right]_{q}(−1)^{k}q^{−(n−1)k+\binom{k}{2}} ∑_{1≤l≤k} \left[{k − 1 \atop l − 1}\right]_{q}(−1)^{l}q^{\binom{l}{2}}a_{l}

= ∑_{1≤l≤n}(−1)^{l}q^{\binom{l}{2}}a_{l} ∑_{l≤k≤n} \left[{n − 1 \atop k − 1}\right]_{q}\left[{k − 1 \atop l − 1}\right]_{q}(−1)^{k}q^{−(n−1)k+\binom{k}{2}} = a_{n},

since for l < n the inner sum is

∑_{l≤k≤n}\left[{n − 1 \atop k − 1}\right]_{q}\left[{k − 1 \atop l − 1}\right]_{q}(−1)^{k}q^{−(n−1)k+\binom{k}{2}}

= (−1)^{l}q^{−(n−1)l+\binom{l}{2}}\left[{n − 1 \atop l − 1}\right]_{q} ∑_{0≤m≤n−l}(−1)^{m}(q^{−(n−l−1)})^{m}q^{\binom{m}{2}}\left[{n − l \atop m}\right]_{q}

= (−1)^{l}q^{−(n−1)l+\binom{l}{2}}\left[{n − 1 \atop l − 1}\right]_{q}(q^{−(n−l−1)}; q)_{n−l} = 0;

here we have used the following version of the q–binomial theorem (see e. g. [1]):

∑_{0≤m≤n}(−1)^{m}z^{m}q^{\binom{m}{2}}\left[{n \atop m}\right]_{q} = (z; q)_{n}.

The other direction is very similar. (For our application “Dilcher” ⇒ “Hernández” we would need only one implication.)

We would like to remark that an alternative formulation can be given in terms of matrices of connection coefficients.

This can be done in terms of the original formulæ (4), but looks much nicer when referring to (5):

Define matrices

T := [\left[{n − 1 \atop k − 1}\right]_{q}(−1)^{k}q^{\binom{k}{2}}]_{n,k} , U := [\left[{n − 1 \atop k − 1}\right]_{q}(−1)^{k}q^{−(n−1)k+\binom{k}{2}}]_{n,k} ,

then

T U = I.

Theorem 2. [q–analogue of Hernández’ formula]

∑_{1≤k≤n} \left[{n \atop k}\right]_{q}(−1)^{k−1}q^{−kn+\binom{k}{2}} ∑_{1≤i_{1}≤i_{2}≤···≤i_{m}=k} \frac{q^{i_{1}}}{1 − q^{i_{1}}} … \frac{q^{i_{m}}}{1 − q^{i_{m}}} = ∑_{1≤k≤n} \frac{q^{k(m−1)}}{(1 − q^{k})^{m}} .

Acknowledgment. The presentation has gained much by the comments of an insightful referee.

References

[1] G. Andrews. The Theory of Partitions, volume 2 of Encyclopedia of Mathematics and its Applications. Addison–Wesley, 1976.

[2] G. Andrews, D. Crippa, and K. Simon. q–series arising from the study of random graphs. SIAM Journal on Discrete Mathematics, 10:41–56, 1997.

[3] K. Dilcher. Some q–series identities related to divisor functions. Discrete Mathematics, 145:83–93, 1995.

[4] P. Flajolet and R. Sedgewick. Mellin transforms and asymptotics: Finite differences and Rice’s integrals. Theoretical Computer Science, 144:101–124, 1995.

[5] I. Gessel and D. Stanton. Applications of q–Lagrange inversion to basic hypergeometric series. Transactions of the American Mathematical Society, 227:173–201, 1983.

[6] V. Hernández. Solution IV of problem 10490 (a reciprocal summation identity). American Mathematical Monthly, 106:589–590, 1999.

[7] J. Riordan. Combinatorial Identities. John Wiley, 1968.

Helmut Prodinger, Centre for Applicable Analysis and Number Theory, Department of Mathematics, University of the Witwatersrand, P. O. Wits, 2050 Johannesburg,

South Africa, email: helmut@gauss.cam.wits.ac.za,
homepage: http://www.wits.ac.za/helmut/index.htm.