Generating asymptotics for factorially divergent sequences Michael - PowerPoint PPT Presentation

Generating asymptotics for factorially divergent sequences Michael Borinsky 1 Humboldt-University Berlin Departments of Physics and Mathematics ALEA in Europe, Vienna 2017 1 borinsky@physik.hu-berlin.de M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 1

Introduction Singularity analysis is a great tool to obtain asymptotic expansions of combinatorial classes. Caveat: Only applicable if the generating function has a non-zero, finite radius of convergence. Topic of this talk: Power series with vanishing radius of convergence and factorial growth. M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 2

Consider the class of power series R [[ x ]] α β ⊂ R [[ x ]] which admit an asymptotic expansion of the form, � � c 1 c 2 f n = α n + β Γ( n + β ) c 0 + α ( n + β ) + α 2 ( n + β )( n + β − 1) + . . . R − 1 � � � c k α n + β − k Γ( n + β − k ) + O α n + β − R Γ( n + β − R ) = k =0 R [[ x ]] α β a linear subspace of R [[ x ]]. Includes power series with non-vanishing radius of convergence: In this case all c k = 0. These power series appear in Graph counting Permutations Perturbation expansions in physics M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 3

Consider a power series f ( x ) ∈ R [[ x ]] α β : R − 1 � � � c k α n + β − k Γ( n + β − k ) + O α n + β − R Γ( n + β − R ) f n = k =0 Interpret the coefficients c k of the asymptotic expansion as a new power series. Definition A maps a power series to its asymptotic expansion: R [[ x ]] α A : → R [[ x ]] β ∞ � c k x k f ( x ) �→ γ ( x ) = k =0 M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 4

Theorem A is a derivation on R [[ x ]] α β : ( A f · g )( x ) = f ( x )( A g )( x ) + ( A f )( x ) g ( x ) ⇒ R [[ x ]] α β is a subring of R [[ x ]]. Proof sketch With h ( x ) = f ( x ) g ( x ), R − 1 R − 1 n − R � � � h n = f n − k g k + f k g n − k + f k g n − k k =0 k =0 k = R � �� � � �� � High order times low order O ( α n Γ( n + β − R )) . � n − R k = R f k g n − k ∈ O ( α n Γ( n + β − R )) follows from the log-convexity of the Γ function. M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 5

Example n =1 n ! x n = � ∞ Set F ( x ) = � ∞ n =1 1 n +1 Γ( n + 1) x n , By definition: F ∈ R [[ x ]] 1 1 and ( A F )( x ) = 1 1 is a ring: F ( x ) 2 ∈ R [[ x ]] 1 Because R [[ x ]] 1 1 Because of the product rule for A : ( A F ( x ) 2 )( x ) = F ( x )( A F )( x ) + ( A F )( x ) F ( x ) = 2 F ( x ) Asymptotic expansion of F ( x ) 2 is given by 2 F ( x ): R − 1 � [ x n ] F ( x ) 2 = c k ( n − k )! + O (( n − R )!) ∀ R ∈ N 0 k =0 where c k = [ x k ]2 F ( x ). M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 6

What happens for composition of power series ∈ R [[ x ]] α β ? Theorem Bender [1975] If | f n | ≤ C n then, for g ∈ R [[ x ]] α β with g 0 = 0: f ◦ g ∈ R [[ x ]] α β ( A f ◦ g )( x ) = f ′ ( g ( x ))( A g )( x ) . Bender considered much more general power series, but this is a direct corollary of his theorem in 1975. M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 7

Example A reducible permutation: An irreducible permutation: 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 A permutation π of [ n ] = { 1 , . . . , n } is called irreducible if there is no m < n such that π ([ m ]) = [ m ]. Set F ( x ) = � ∞ n =1 n ! x n - the OGF of all permutations. The OGF of irreducible permutations I fulfills 1 I ( x ) = 1 − 1 + F ( x ) . M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 8

∞ 1 � n ! x n . I ( x ) = 1 − F ( x ) = 1 + F ( x ) n =1 By definition: F ∈ R [[ x ]] 1 1 and ( A F )( x ) = 1. 1 1+ x is analytic at the origin, therefore by the chain rule � � �� 1 1 ( A I )( x ) = A 1 − ( x ) = (1 + F ( x )) 2 1 + F ( x ) Theorem Comtet [1972] Therefore the asymptotic expansion of the coefficients of I ( x ) is R − 1 � [ x n ] I ( x ) = c k ( n − k )! + O (( n − R )!) ∀ R ∈ N 0 , k =0 1 where c k = [ x k ] (1+ F ( x )) 2 . M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 9

This chain rule can easily be generalized to multivalued analytic functions: Theorem MB [2016] More general: For f ∈ R { y 1 , . . . , y L } and g 1 , . . . , g L ∈ x R [[ x ]] α β : L ∂ f � ( A ( f ( g 1 , . . . , g L ))( x ) = ∂ g l ( g 1 , . . . , g L )( A α β g l )( x ) . l =1 M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 10

What happens if f is not an analytic function? A fulfills a general ‘chain rule’: Theorem MB [2016] If f , g ∈ R [[ x ]] α β with g 0 = 0 and g 1 = 1, then f ◦ g ∈ R [[ x ]] α β and � � β x g ( x ) − x α xg ( x ) ( A f )( g ( x )) ( A f ◦ g )( x ) = f ′ ( g ( x ))( A g )( x ) + e g ( x ) ⇒ R [[ x ]] α β is closed under composition and inversion. ⇒ We can solve for asymptotics of implicitly defined power series. M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 11

Example: Simple permutations A non-simple permutation: A simple permutation: 8 8 7 7 6 6 5 5 4 4 3 3 2 2 1 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 A permutation π of [ n ] = { 1 , . . . , n } is called simple if there is no (non-trivial) interval [ i , j ] = { i , . . . , j } such that π ([ i , j ]) is another interval. The OGF S ( x ) of simple permutations fulfills F ( x ) − F ( x ) 2 = x + S ( F ( x )) , 1 + F ( x ) with F ( x ) = � ∞ n =1 n ! x n [Albert, Klazar, and Atkinson, 2003]. M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 12

F ( x ) − F ( x ) 2 = x + S ( F ( x )) . 1 + F ( x ) By definition: F ∈ R [[ x ]] 1 1 and ( A F )( x ) = 1. Extract asymptotics by applying the A -derivative: � F ( x ) − F ( x ) 2 � A = A ( x + S ( F ( x ))) . 1 + F ( x ) Apply chain rule on both sides 1 − 2 F ( x ) − F ( x ) 2 ( A F )( x ) = S ′ ( F ( x ))( A F )( x ) (1 + F ( x )) 2 � � 1 x F ( x ) − x xF ( x ) ( A S )( F ( x )) , + e F ( x ) which can be solved for ( A S )( x ). M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 13

After simplifications: 2+(1+ x ) S ( x ) 1 − x − (1 + x ) S ( x ) x 2 − 1 1 − x − (1+ x ) S ( x ) x ( A S )( x ) = e x 1 + x 1 + (1 + x ) S ( x ) x 2 We get the full asymptotic expansion for S : R − 1 � [ x n ] S ( x ) = c k ( n − k )! + O (( n − R )!) ∀ R ∈ N 0 k =0 where c k = [ x k ]( A S )( x ). � � 1 − 4 2 40 [ x n ] S ( x ) = e − 2 n ! n + n ( n − 1) − 3 n ( n − 1)( n − 2) + . . . , the first three coefficients have been obtained by Albert, Klazar, and Atkinson [2003]. M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 14

Meta asymptotics 2+(1+ x ) S ( x ) 1 − x − (1 + x ) S ( x ) x 2 − 1 1 − x − (1+ x ) S ( x ) x ( A S )( x ) = e := g ( x , S ( x )) x 1 + x 1 + (1 + x ) S ( x ) x 2 g ( x , S ( x )) is an analytic function in S ( x ): Because of the chain rule for analytic functions, ( A ( A S ))( x ) = ∂ g ( x , S ) ( A S )( x ) , ∂ S we obtain the asymptotics of the asymptotic expansion . M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 15

2+(1+ x ) S 1 − x − (1 + x ) S x 2 1 − 1 − x − (1+ x ) S x g ( x , S ) = e x 1 + (1 + x ) S 1 + x x 2 This way we can obtain the GF for meta asymptotics : ∞ t k ( A k S )( x ) � = q − 1 ( t + q ( S ( x ))) , f ( t , x ) = k ! k =0 � S g ( x , S ′ ) and q − 1 ( q ( S )) = S . dS ′ where q ( S ) = 0 [ t k ] f ( t , x ) is the GF of the k -th order asymptotics of S . Using this information to resum such a series leads to the theory of resurgence. M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 16

Conclusions R [[ x ]] α β forms a subring of R [[ x ]] closed under mutliplication, composition and inversion . A is a derivation on R [[ x ]] α β which can be used to obtain asymptotic expansions of implicitly defined power series . Closure properties under asymptotic derivative A . M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 17

MH Albert, M Klazar, and MD Atkinson. The enumeration of simple permutations. 2003. Edward A Bender. An asymptotic expansion for the coefficients of some formal power series. Journal of the London Mathematical Society , 2(3):451–458, 1975. Louis Comtet. Sur les coefficients de l’inverse de la s´ erie formelle � n ! t n . CR Acad. Sci. Paris, Ser. A , 275(1):972, 1972. MB. Generating asymptotics for factorially divergent sequences. arXiv preprint arXiv:1603.01236 , 2016. M. Borinsky (HU Berlin) Generating asymptotics for factorially divergent sequences 17