By Carl Faith

E-book by means of religion, Carl

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**Extra resources for Algebra II. Ring Theory: Ring Theory**

**Example text**

However, if we can ﬁnd a closed form for its generating function, we have all the information about its coeﬃcients as well. In his paper, Gessel [7] provided a simple way using symmetric functions to obtain the generating functions of certain CHAPTER 5. D-FINITENESS: SCALAR AND KRONECKER PRODUCTS 46 combinatorial structures, such as partitions of multisets and nonnegative integer matrices. Chapter 6 Reduced Kronecker product In 1938, Murnaghan [17] introduced the concept of reduced notation for Schur functions.

Moreover, if we consider the ring Q[[t, p1 , p2 , . . , pn , . ]], then a symmetric series f ∈ Q[[t, p1 , p2 , . . , pn , . ]] is D-ﬁnite if it is D-ﬁnite with respect to t and the pi ’s. 5. 5). If we set pk = 0 k≥1 k for k > n0 for some n0 ∈ N, we get h|n0 = exp (n ) 0 ∑ pk k=1 k which is clearly D-ﬁnite with respect the pi ’s. Notice that F = exp(h) is D-ﬁnite with respect the hi ’s, but we cannot say the same with respect to the pi ’s, because ∑ pk , F = exp exp k k≥1 CHAPTER 4.

Then it remains for us to show that Dh h(t)g is D-ﬁnite. Indeed: Dh h(t)g = ∑ tn Dh (hn g), n = ∑ tn (hn g)[x + 1], since Dh (g) = g[x + 1], see [24] n = ∑ n tn hn [x + 1]g[x + 1], CHAPTER 6. REDUCED KRONECKER PRODUCT 55 where g[x + 1] is a D-ﬁnite polynomial and ∑ n t hn [x + 1] = exp n ( ) ∑ tk (pk + 1) k = exp ( ∑ t k pk k k k ) , ) ( ∑ tk exp , k k 1 , which is D-ﬁnite. 3 we have that ⟨hf, h(t)g⟩ is D-ﬁnite as wanted. ♣ Notice that this proof holds also if f and g involve a ﬁnite number of pi ’s but are not necessarily polynomials.