Normally ordered forms for powers of differential operators Emmanuel Briand Mercedes Rosas

Differential operators multiplication by derivation w.r.t. As operators, which just means

Key for the proofs Leibniz rule in tensor language:

arXiv:1811.00857

Samuel Lopes

1

Normal ordering

Multiply everything acts on factor 1 on factor 0

Not normally ordered

Normally ordered (all to the right).

(Leibniz' rule)

The coefficients are the Partition Stirling Numbers ("2nd kind") that count Set Partitions / increasing forests with linear trees.

1 3 2

Increasing forests with linear trees:

1 3 2 1

1 2 3

1

0

The Cayley monomial of an increasing forest

Classical: Normally ordered form of Set partitions:

2

Factors:

0

1 3 2 1

3

2

3

1 3 2

1

2

1

1 3 2 1 2 3

2 3

4

8

6

5

2

3

12

9

increasing labels 10

7

11

Replace each node with k sons with

Normally ordered form of Theorem (Bergeron‐Reutenauer 1987) The normally ordered form Increasing of is the sum of all forests: Cayley monomials of increasing forests on n nodes.

1 1 2 3

1

3

2

3

1

2

1 2 3

2 3

for 3 trees in total.

1 3

2

The Cayley monomial of a pair of increasing forests

Normally ordered form of Theorem: The normally ordered form of is the sum of the Cayley monomials associated to all d‐tuples of increasing forests on n nodes. 1 1 2 3

1

1

3

2

1 3

2

1

2 3

1 2 3

4

1 2

3

&

2 5

3 5

4

6

3

2 6

Superpose 1

1 2 3 1

3

3

2

2 1 3

1

4 2 2

5

6

Replace each node with k outgoing edges with

3 1 2 3

for 3 trees in total.

1 2

3

Corollary: new formulas for Generalized Stirling Numbers The generalized Stirling numbers

Example:

Theorem:

Sum over all triangular arrays with rows and columns, and total sum

are the coefficients in the normally ordered form of

(falling factorial)

Sum over all arrays with sum

E. Briand y M. Rosas are partially supported by MTM2016-75024-P and FEDER, and Junta de Andalucia under grants P12-FQM-2696 and FQM-333. S. Lopes is partially supported by CMUP (UID/MAT/00144/2013), which is funded by FCT (Portugal) with national (MEC) and European structural funds (FEDER), under the partnership agreement PT2020.