On q-Analog of Wolstenholme Type Congruences for Multiple Harmonic Sums

Multiple harmonic sums are iterated generalizations of harmonic sums. Recently Dilcher has considered congruences involving q-analogs of these sums in depth one. In this paper we shall study the homogeneous case for arbitrary depth by using generating functions and shuffle relations of the q-analog of multiple harmonic sums. At the end, we also consider some non-homogeneous cases.


Introduction.
In [8] Shi and Pan extended Andrews' result [1] on the q-analog of Wolstenholme Theorem to the following two cases: for all prime p ≥ 5 where [n] q = (1 − q n )/(1 − q) for any n ∈ N and q = 1. This type of congruences is considered in the polynomial ring Z[q] throughout this paper.
Notice that the modulus [p] q is an irreducible polynomial in q when p is a prime. In [3] Dilcher generalized the above two congruences further to sums of the form p−1 j=1 1 [j] n q and p−1 j=1 q n [j] n q for all positive integers n in terms of certain determinants of binomial coefficients. However, his modulus is always [p] q . He also expressed these congruences using Bernoulli numbers, Bernoulli numbers of the second kind, and Stirling numbers of the first kind, which we briefly recall now.
The well-known Bernoulli numbers are defined by the following generating series: x On the other hand, the Bernoulli numbers of the second kind are defined by the power series (cf. [7, p. 114]).
This is a little different from the definition ofb n in [3], which is changed to b n later in the same paper. Finally, the Stirling numbers of the first kind s(n, j) are defined by We will need the following easy generalization of this theorem.
Theorem 1.2. If p > 3 is a prime, then for all integers n > t ≥ 1 we have Moreover, Proof. If t > 1 it is clear that So (4) follows from Theorem 1.1 immediately. Congruence (5) is a variation of [3, (5.11)].
All of the sums in Theorem 1.1 and 1.2 are special cases of the q-analog of multiple harmonic sums. The congruence properties of the classical multiple harmonic sums (MHS for short) are systematically investigated in [10]. In this paper we shall study their q-analogs which are natural generalizations of the congruences obtained by Shi and Pan [8] and Dilcher [3].
In this paper we mainly consider q-MHS with the trivial modifier. By convention we set H To save space, for an ordered set (e 1 , . . . , e t ) we denote by {e 1 , . . . , e t } d the ordered set formed by repeating (e 1 , . . . , e t ) d times. For example H q ({s} ℓ ; n) will be called a homogeneous sum.
Throughout the paper, we use short-hand H q (s) to denote H q (s; p − 1) for some fixed prime p.

Homogeneous q-MHS.
It is extremely beneficial to study the so-called stuffle (or quasi-shuffle) relations among MHS (see, for e.g., [10]). The same mechanism works equally well for q-MHS.
Recall that for any two ordered sets (r 1 , . . . , r t ) and (r t+1 , . . . , r n ) the shuffle operation is defined by Fix a positive integer s. For any k = 1, . . . , ℓ − 1, we have by stuffle relation Theorem 2.1. Let s be a positive integer and let η s = exp(2πi/s) be the sth primitive root of unity. Then Proof. Let ζ = exp(2πi/p) be the primitive pth root of unity and set It is easy to see that H * q (n) ≡ P n (mod [p] q ). By using partial fractions Dilcher [4, (4.2)] obtained essentially the following generating function of P n : Let a ℓ = H * q {s} ℓ for all ℓ ≥ 0. Let w(x) = ∞ ℓ=0 a ℓ x ℓ be its the generating function. By (7) we get Differentiating both sides and changing index ℓ → ℓ + 1 we get modulo Here η s = exp(2πi/s) is the sth primitive root of unity. Thus Therefore by comparing the constant term we get as desired.

Corollary 2.2. For all positive integer ℓ < p we have
Proof. By the theorem we get The corollary follows immediately.

Corollary 2.3. For every positive integer ℓ < p we have
where F 2,ℓ (p) is a monic polynomial in p of degree ℓ.

Proof. By Theorem 2.1 we have modulo [p]
which easily yields In the first sum above if j + k = ℓ + 1 and 1 ≤ j, k < p/2 then we may assume j > ℓ/2. Then (ℓ + 1)! p ℓ+1 is a factor of (2j + 1)! p 2j+1 as a polynomial of p, so is ℓ! p−1 ℓ . Similarly we can see that ℓ! p−1 ℓ is a factor of the second sum.
In order to determine the leading coefficient we set Hence j+k=ℓ 0≤j,k<p/2 This finishes the proof of the corollary.
where F 3,ℓ (p) is a monic polynomial in p of degree 2ℓ − 1 if ℓ is odd and of degree 2ℓ if ℓ is even.
Proof. Let η = exp(2πi/3). Then η 2 + η + 1 = 0. By Theorem 2.1 we have We now use two ways to expand this. Set y = 3 √ −x. First, the product on the right hand side of (11) can be expressed as Thus for ℓ > 0 we get Note that if ℓ is odd then the degree of the polynomial is reduced to 3ℓ − 1 with leading coefficient given by Now to prove ℓ! p ℓ is a factor we use the following expansion of (11): Thus Notice that j + k + n = 3ℓ + 3 implies one of the indices, say j, is at least ℓ + 1. Then clearly p j contains ℓ! p ℓ as a factor, therefore so does H * q {3} ℓ (mod [p] q ). This completes the proof of the corollary.
In this section we consider some non-homogeneous q-MHS of depth two with modifiers of special type.
Proof. By definition and substitution i → p − i and j → p − j we have By shuffle relation we have Together with (12) this yields Our theorem follows from (4) quickly.
In the study of q-multiple zeta functions the following function appears naturally (see [9, (47)] or [2, Theorem 1]): is the q-Riemann zeta value defined by Kaneko et al. in [5]. Using the results we have obtained so far in this paper we discover a congruence related to the partial sums of ϕ q (2).

A congruence of Lehmer type
Instead of the harmonic sums up to (p − 1)-st term Lehmer also studied the following type of congruence (see [6]): for every odd prime p where q p (2) = (2 p−1 − 1)/p is the Fermat quotient. It is also easy to see that for every positive integer n and prime p > 2n + 1 As a q-analog of the above we have To conclude the paper we remark that the congruence for general q-MHS should involve some type of q-analog of Bernoulli numbers and Euler numbers similar to the classical cases treated in [10]. We hope to return to this theme in the future.