Séminaire Lotharingien de Combinatoire 53 (2005), Article B53c

COMPUTING POWERS

OF TWO GENERALIZATIONS OF THE LOGARITHM

Wadim Zudilin^{‡} (Moscow)

Abstract. We prove multiple-series representations for positive integer powers of the series

L(z; α) = ∑_{n=1}^{∞} \frac{z^{n}}{n + α} , |z| < 1, α ⩾ 0, and ℓ_{q}(z) = ∑_{n=1}^{∞} \frac{z^{n}q^{n}}{1 − q^{n}} , |z| ⩽ 1, |q| < 1.

The results generalize a known formula for powers of the series for the ordinary logarithm − log(1 − z) = L(z; 0).

The series for the ordinary logarithm,

Li_{1}(z) = − log(1 − z) = ∑_{n=1}^{∞} \frac{z^{n}}{n} ,  |z| < 1,  (1)

admits the following formula

Li_{1}(z)^{l} = l! ∑_{n_{l}>n_{l−1}>···>n_{1}⩾1} \frac{z^{n_{l}}}{n_{1}n_{2} ⋯ n_{l}} .  (2)

It plays an important rôle in evaluating irrationality and transcendence measures for values of the logarithm at rational points (cf. [Ne]). On the other hand, information on the arithmetic nature of values of the following two generalizations of the series (1) is available only in particular cases at this moment. The first generalization, the α-shift of the logarithm for α ⩾ 0, is defined by the series

L(z) = L(z; α) = ∑_{n=1}^{∞} \frac{z^{n}}{n + α}, |z| < 1.

The second generalization, known as a q-extension of the logarithm, is given by the formula

ℓ_{q}(z) = ∑_{n=1}^{∞} \frac{z^{n}q^{n}}{1 − q^{n}} = ∑_{n=1}^{∞} \frac{z^{n}}{p^{n} − 1}, |z| ⩽ 1, |q| < 1, p = q^{−1}.

‡The work is partially supported by grant no. 03-01-00359 of the Russian Foundation for Basic Research.

Then one has

lim_{\begin{matrix}q→1 \\ |q|<1 \end{matrix}}(1 − q)ℓ_{q}(z) = Li_{1}(z) = L(z; 0) for |z| < 1.

The aim of this note is to provide generalizations of the formula (2) for both L(z; α) (Theorem 1) and ℓ_{q}(z) (Theorem 2), which might become useful in the arithmetic study of values of the series.

Theorem 1. For each l = 0, 1, 2, …, the following identity holds:

L(z)^{l} = l!L_{l}(z), l = 0, 1, 2, …,

where

L_{l}(z) = ∑_{n_{l}>···>n_{2}>n_{1}⩾1} \frac{z^{n_{l}}}{(n_{1} + α) ⋯ (n_{l} + lα)}, l = 1, 2, …,  L_{0}(z) = 1.

Proof. For given l, we have

z^{1−lα} \frac{d}{dz} (z^{lα}L_{l}(z)) = z^{1−lα} \frac{d}{dz} ∑_{n_{l}>···>n_{1}⩾1} \frac{z^{n_{l}+lα}}{(n_{1} + α) ⋯ (n_{l−1} + (l − 1)α)(n_{l} + lα)}

= ∑_{n_{l−1}>···>n_{1}⩾1} \frac{1}{(n_{1} + α) ⋯ (n_{l−1} + (l − 1)α)} ∑_{n_{l}=n_{l−1}+1}^{∞} z^{n_{l}}

= \frac{z}{1 − z} ∑_{n_{l−1}>···>n_{1}⩾1} \frac{z^{n_{l−1}}}{(n_{1} + α) ⋯ (n_{l−1} + (l − 1)α)}

= \frac{z}{1 − z} L_{l−1}(z).

On the other hand,

z^{1−lα} \frac{d}{dz} (z^{lα}L_{l}(z)) = z^{1−lα}(lαz^{lα−1}L_{l}(z) + z^{lα} \frac{d}{dz} L_{l}(z))

= lαL_{l}(z) + z \frac{d}{dz} L_{l}(z).

Therefore,

z \frac{d}{dz} L_{l}(z) = −lαL_{l}(z) + \frac{z}{1 − z} L_{l−1}(z), l = 1, 2, ….

If we define

L̃_{l}(z) = \frac{1}{l!}L(z)^{l} = \frac{1}{l!}(∑_{n=1}^{∞} \frac{z^{n}}{n + α})^{l},

COMPUTING POWERS OF TWO GENERALIZATIONS OF THE LOGARITHM 3

then

z \frac{d}{dz} L̃_{l}(z) = \frac{1}{l!}z \frac{d}{dz} L(z)^{l} = \frac{1}{(l − 1)!}L(z)^{l−1} · z \frac{d}{dz} L(z)

= \frac{1}{(l − 1)!}L(z)^{l−1}(−αL(z) + \frac{z}{1 − z} ) = −αlL̃_{l}(z) + \frac{z}{1 − z} L̃_{l−1}(z).

Since L_{0}(z) = L̃_{0}(z) = 1 and L_{l}(z) = (z/(1+α))^{l}/l!+O(z^{l+1}), L̃_{l}(z) = (z/(1+α))^{l}/l!+ O(z^{l+1}), we obtain the desired result. □ Theorem 2. For each l = 1, 2, …,

ℓ_{q}(z)^{l} = ∑_{n_{l}>n_{l−1}>···>n_{1}⩾1} \frac{z^{n_{l}}q^{n_{l}}Φ_{l}(q^{n_{1}}, q^{n_{2}−n_{1}}, …, q^{n_{l}−n_{l−1}})}{(1 − q^{n_{1}})(1 − q^{n_{2}}) ⋯ (1 − q^{n_{l}})}
= ∑_{n_{l}>n_{l−1}>···>n_{1}⩾1} \frac{z^{n_{l}}Φ_{l}(p^{n_{1}}, p^{n_{2}−n_{1}}, …, p^{n_{l}−n_{l−1}})}{(p^{n_{1}} − 1)(p^{n_{2}} − 1) ⋯ (p^{n_{l}} − 1)},  (3)

where Φ_{l}(x_{1}, …, x_{l}) is the polynomial

Φ_{l}(x_{1}, …, x_{l}) = (x_{1}^{l−1} + x_{1}^{l−2} + ⋯ + x_{1} + 1)(x_{2}^{l−2} + ⋯ + x_{2} + 1) ⋯ (x_{l−1} + 1)
= ∏_{j=1}^{l} \frac{x_{j}^{l+1−j} − 1}{x_{j} − 1}.  (4)

In particular, one has

\begin{matrix}ℓ_{1/p}(z)^{2} = ∑_{n_{2}>n_{1}⩾1} \frac{z^{n_{2}}(p^{n_{1}} + 1)}{(p^{n_{1}} − 1)(p^{n_{2}} − 1)}, \\ ℓ_{1/p}(z)^{3} = \frac{1}{2} ∑_{n_{3}>n_{2}>n_{1}⩾1} \frac{z^{n_{3}}(p^{2n_{1}} + p^{n_{1}} + 1)(p^{n_{2}−n_{1}} + 1)}{(p^{n_{1}} − 1)(p^{n_{2}} − 1)(p^{n_{3}} − 1)}. \end{matrix}

The proof requires two auxiliary identities. Lemma 1. For l = 2, 3, …, the equality

\frac{1}{(x_{1} − 1)(x_{1}x_{2} − 1) ⋯ (x_{1}x_{2} ⋯ x_{l−1} − 1) · (x − 1)}

= ∑_{j=1}^{l} \frac{x_{1} … x_{j−1}}{∏_{k=1}^{j−1}(x_{1} ⋯ x_{k} − 1) · ∏_{k=j−1}^{l−1}(x_{1} ⋯ x_{k}x − 1)}

holds identically in the variables x_{1}, …, x_{l−1} and x. (Empty products should be replaced by 1.)

The proof exploits a simple inductive argument, and therefore is omitted.

Lemma 2. For each l = 1, 2, …, the following identity holds:

\frac{1}{(x_{1} − 1)(x_{2} − 1) ⋯ (x_{l} − 1)}

= \frac{1}{l!} ∑_{σ∈𝔖_{l}} \frac{Φ_{l}(x_{σ(1)}, …, x_{σ(l)})}{(x_{σ(1)} − 1)(x_{σ(1)}x_{σ(2)} − 1) ⋯ (x_{σ(1)}x_{σ(2)} ⋯ x_{σ(l)} − 1)},  (5)

where

Φ_{l}(x_{1}, …, x_{l}) = ∑_{\begin{matrix}k_{i}=0 \\ i=1,...,l \end{matrix}}^{l−i} x_{1}^{k_{1}}x_{2}^{k_{2}} ⋯ x_{l}^{k_{l}}  (6)

is the polynomial defined in (4) and 𝔖_{l} denotes the set of all permutations of {1, 2, …, l}.

Proof. We apply induction on l. If l = 1, then both sides of (5) are simply 1/(x_{1} − 1). Suppose that l > 1 and that we have proved identity (5) for l replaced by l − 1. Then applying Lemma 1 (with x = x_{l}) we obtain

\frac{1}{(x_{1} − 1)(x_{2} − 1) ⋯ (x_{l−1} − 1) · (x_{l} − 1)}

= \frac{1}{(l − 1)!} ∑_{σ∈𝔖_{l−1}} \frac{Φ_{l−1}(x_{σ(1)}, …, x_{σ(l−1)})}{(x_{σ(1)} − 1)(x_{σ(1)}x_{σ(2)} − 1) ⋯ (x_{σ(1)} ⋯ x_{σ(l−1)} − 1) · (x_{l} − 1)}

= \frac{1}{(l − 1)!} ∑_{σ∈𝔖_{l−1}} ∑_{j=1}^{l} \frac{Φ_{l−1}(x_{σ(1)}, …, x_{σ(l−1)})}{∏_{k=1}^{j−1}(x_{σ(1)} ⋯ x_{σ(k)} − 1)}

× \frac{x_{σ(1)} ⋯ x_{σ(j−1)}}{∏_{k=j−1}^{l−1}(x_{σ(1)} ⋯ x_{σ(j−1)}x_{l}x_{σ(j)} ⋯ x_{σ(k)} − 1)}

= \frac{1}{(l − 1)!} ∑_{τ∈𝔖_{l}} \frac{Φ_{l−1}(x_{τ(1)}, …, { \atop x_{τ(j)}}, …, x_{τ(l)}) · x_{τ(1)} ⋯ x_{τ(j−1)}}{∏_{k=1}^{l}(x_{τ(1)} ⋯ x_{τ(k)} − 1)} ,  (7)

where in the latter sum j abbreviates τ^{−1}(l), that is

τ = τ(j, σ) ∈ 𝔖_{l}: (1, 2, …, l − 1, l) ↦ (σ(1), …, σ(j − 1), l, σ(j), …, σ(l − 1)),

and the notation x̂ means omitting the corresponding parameter. Since the product on the left-hand side of (7) is symmetric in x_{1}, x_{2}, …, x_{l}, we may replace x_{l} by x_{i} with i ≠ l, to deduce an identity similar to but different from (7). If we now average the results over all i = 1, …, l, then we obtain

\frac{1}{(x_{1} − 1)(x_{2} − 1) ⋯ (x_{l−1} − 1)(x_{l} − 1)}

= \frac{1}{(l − 1)!} · \frac{1}{l} ∑_{j=1}^{l} ∑_{τ∈𝔖_{l}} \frac{Φ_{l−1}(x_{τ(1)}, …, { \atop x_{τ(j)}}, …, x_{τ(l)}) · x_{τ(1)} ⋯ x_{τ(j−1)}}{∏_{k=1}^{l}(x_{τ(1)} ⋯ x_{τ(k)} − 1)}

= \frac{1}{l!} ∑_{τ∈𝔖_{l}} \frac{Φ̃_{l}(x_{τ(1)}, …, x_{τ(l)})}{∏_{k=1}^{l}(x_{τ(1)} ⋯ x_{τ(k)} − 1)},

where we set

Φ̃_{l}(x_{1}, …, x_{l}) = ∑_{j=1}^{l} Φ_{l−1}(x_{1}, …, x̂_{j}, …, x_{l}) · x_{1}x_{2} ⋯ x_{j−1}.  (8)

It remains to verify that the polynomials Φ̃_{l} in (8) and Φ_{l} in (6) coincide:

Φ̃_{l}(x_{1}, …, x_{l}) = ∑_{j=1}^{l} ∑_{\begin{matrix}k_{i}=0 \\ i=1,...,l−1 \end{matrix}}^{l−1−i} x_{1}^{k_{1}+1} ⋯ x_{j−1}^{k_{j−1}+1}x_{j+1}^{k_{j}} ⋯ x_{l}^{k_{l−1}}

= ∑_{j=1}^{l} ∑_{\begin{matrix}k′_{i}=1 \\ i=1,...,j−1 \end{matrix}}^{l−i} x_{1}^{k′_{1}} ⋯ x_{j−1}^{k′_{j−1}} ∑_{\begin{matrix}k′_{i}=0 \\ i=j+1,...,l \end{matrix}}^{l−i} x_{j+1}^{k′_{j+1}} ⋯ x_{l}^{k′_{l}}

= ∑_{\begin{matrix}k′_{i}=0 \\ i=1,...,l \end{matrix}}^{l−i} x_{1}^{k′_{1}}x_{2}^{k′_{2}} ⋯ x_{l}^{k′_{l}} = Φ_{l}(x_{1}, …, x_{l}),

where we use the bijection

(j, k_{1}, …, k_{l−1}) ↦ (k′_{1}, k′_{2}, …, k′_{l}) = (k_{1} + 1, …, k_{j−1} + 1, 0, k_{j}, …, k_{l−1}).

This completes our proof of the lemma. □

Remark. The identity of Lemma 2 belongs to a family of identities in the style of Littlewood [Li], p. 85:

∏_{j=1}^{l} \frac{1}{x_{j}} = ∑_{σ∈𝔖_{l}} \frac{1}{x_{σ(1)}(x_{σ(1)} + x_{σ(2)}) ⋯ (x_{σ(1)} + x_{σ(2)} + ⋯ + x_{σ(l)})}.

Identities of similar type and their applications may be found in [Me], Section 10.9, Appendix A.23. Lassalle in [La] gives formulae, where monomial symmetric functions are specialized; taking as variables the parts of the partition, which indexes the function (i.e., replacing x_{j} by q^{j−1} for j = 1, …, l), one obtains curious generalizations of Littlewood-type identities.

Proof of Theorem 2. From Lemma 2 we deduce that

ℓ_{q}(z)^{l} = ( ∑_{m=1}^{∞} \frac{z^{m}}{p^{m} − 1})^{l} = ∑_{\begin{matrix}m_{i}=1 \\ i=1,...,l \end{matrix}}^{∞} \frac{z^{m_{1}+···+m_{l}}}{(p^{m_{1}} − 1) ⋯ (p^{m_{l}} − 1)}

= \frac{1}{l!} ∑_{\begin{matrix}m_{i}=1 \\ i=1,...,l \end{matrix}}^{∞} ∑_{σ∈𝔖_{l}} \frac{Φ_{l}(x_{σ(1)}, …, x_{σ(l)})}{∏_{k=1}^{l}(x_{σ(1)} ⋯ x_{σ(k)} − 1)}|_{x_{i}=p^{m_{i}} , i=1,...,l}

= ∑_{\begin{matrix}m_{i}=1 \\ i=1,...,l \end{matrix}}^{∞} \frac{Φ_{l}(x_{1}, …, x_{l})}{∏_{k=1}^{l}(x_{1} ⋯ x_{k} − 1)}|_{x_{i}=p^{m_{i}} , i=1,...,l},

and after the change n_{i} = m_{1} + ⋯ + m_{i}, i = 1, …, l, we arrive at the claimed formula (3). □

Acknowledgements. I thank C. Krattenthaler, J. Sondow and the two anonymous referees of the journal for several suggestions.

References

[La] M. Lassalle, Une q-spécialisation pour les fonctions symétriques monomiales, Adv. Math. 162 (2001), no. 2, 217–242.

[Li] D. E. Littlewood, The theory of group characters and matrix representations of groups, 2nd ed., Oxford Univ. Press, Oxford, 1950.

[Me] M. L. Mehta, Random matrices, 2nd ed., Academic Press, Boston, MA, 1991.

[Ne] Yu. V. Nesterenko, On an identity of Mahler, preprint (2003). (Russian)

Department of Mechanics and Mathematics Moscow Lomonosov State University Vorobiovy Gory, GSP-2 119992 Moscow, RUSSIA

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