By Mariolys Rivas

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**Example text**

1 λ C k,l = (l − k + m + 1) for k > m + 1. 2 1 (l + k − m + 2) for m ≥ k − 1. 2 Algorithm. Algorithm. 1: Coeﬃcients for two-row shape Schur functions Some work has been done to characterize the coeﬃcients that appear in the Kronecker product of reduced Schur functions indexed by two-row partitions. 3 (Two-row shape partitions, Scharf, Thibon and Wybourne [20]). For k ≥ l ﬁxed: sk[n] ∗ sl[n] = ∑ λ C k, l sλ[n] = λ p l ∑ ∑ sk−p sl−p sp−q . p=0 q=0 Scharf, Thibon and Wybourne gave explicit formulas and algorithms to calculate the coeﬃcients that appear in this case by noticing that λ can have at most three rows, which results from the Littlewood-Richardson multiplication rule of Schur functions.

Two years later this concept was generalized to the multivariate case by Zeilberger [25]. And after ten years, Gessel [7] developed a D-ﬁniteness theory for symmetric series. In this chapter we recall all these deﬁnitions and some important results concerning the closure properties of D-ﬁniteness with respect to scalar and inner products. 1 Deﬁnitions Consider any ﬁeld Q with characteristic 0 and Q[[x]] the ring of formal power series in x. We say that a power series F ∈ Q[[x]] is D-ﬁnite (diﬀerentiably ﬁnite) in a variable x if F and all of its derivatives F (n) = dn F dxn (n ≥ 1) span a ﬁnite-dimensional vector subspace of Q[[x]] over the ﬁeld Q(x) of rational functions.

2. If n = 10, then s4 + s4,2,1 = s6,5 + s2,4,3,1 , where (6, 5) is a partition, but (2, 4, 3, 1) is not. By applying the Jacobi-Trudi identity we obtain that s2,4,2,1 = −s3,3,3,1 . Thus s4 + s4,2,1 = s6,5 − s3,3,3,1 . Littlewood found the following general formula for the Kronecker product of two reduced Schur functions. CHAPTER 6. 2 (Littlewood[13]). For two ﬁxed partitions λ, µ with |λ| = |µ| and a ﬁxed nonnegative integer n ≥ 0: sλ[n] ∗ sµ[n] = ∑ sλ/αγ sµ/βγ (sα ∗ sβ ). α,β,γ where αγ = (α1 + γ1 , α2 + γ2 , .

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