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Additional info for D-finite symmetric functions

Example text

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.