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By Mariolys Rivas

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1. i i i≥1 CHAPTER 6. REDUCED KRONECKER PRODUCT Using Maple, we get that each term of the form L = exp 57 ( pk ) k prkk ∗ exp ( pk ) k pskk satisfies the differential equation: (krk sk − pk )L + ( ∂2L kp2k 2 ∂pk + (kpk − p2k − krk pk − ksk pk ) ∂L ∂pk ) = 0. 1 satisfies these equations as well. (ri )t (si )t r +s −t i i pi i i i , t ! i t i ti ≤ri ti ≤si i where (ri )ti represents the descending factorial. 3, for which we simply expand the result in terms of the power sum basis and verify that all the coefficients are 0.

Pn ]]. 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.

When F is the family of all partitions, we have: ∑ sλ/µ = h[e1 + e2 ] λ ∑ sµ/λ λ which is D-finite, since h[e1 + e2 ] is D-finite and the second sum over λ is a finite sum of Schur functions (λ is bounded by µ) so it is a polynomial in the pi ’s. 2. Let r be a fixed integer, for F = {λ : λ has at most r rows}(known as the partitions of bounded height) and a fixed partition µ, Gessel proved that the symmetric series Br (µ) = ∑ sλ/µ λ∈F is D-finite. In 1968, Gordon and Houten [9] and Bender and Knuth [3] had given a formula for this series, and two years later Gordon [8] published a simplification of this formula.

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