INVERSE FACTORIAL SERIES: A LITTLE KNOWN TOOL FOR THE SUMMATION OF - PDF document

INVERSE FACTORIAL SERIES: A LITTLE KNOWN TOOL FOR THE SUMMATION OF DIVERGENT SERIES Ernst Joachim Weniger Theoretical Chemistry University of Regensburg, Germany joachim.weniger@chemie.uni-regensburg.de Approximation and Extrapolation of Convergent and Divergent Sequences and Series CIRM Luminy, France 28th September – 2nd October 2009 1

“Rediscovery” of Factorial Series ⊲ In 1985/6, I tried to understand Levin’s se- quence transformation whose input data are not only sequence elements { s n } ∞ n =0 , but also explicit remainder estimates { ω n } ∞ n =0 . ⊲ Levin’s sequence transformation can be con- structed via the model sequence k − 1 c j s n − s � = ( n + β ) j , n ∈ N 0 , β > 0 . ω n j =0 ⇒ The weighted difference operator ∆ k ( n + β ) k − 1 (acting on n ) produces an explicit expression. ⊲ Replacing powers ( n + β ) j by Pochhammer symbols ( n + β ) j yields the model sequence k − 1 c j s n − s � = n ∈ N 0 , β > 0 . , ( n + β ) j ω n j =0 ⇒ Here, ∆ k ( n + β ) k − 1 does the job and yields an expression for a sequence transformation. ⊲ What are series involving inverse Pochham- mer symbols? Are they something useful? 2

Definition of Factorial Series ⊲ Let Ω: C → C be a function which vanishes as z → + ∞ . A factorial series for Ω( z ) is an expansion whose z -dependence occurs in Pochhammer symbols in the denominator: a 1 1! a 2 2! Ω( z ) = a 0 + z ( z + 1) + z ( z + 1)( z + 2) + · · · z ∞ a ν ν ! � = . ( z ) ν +1 ν =0 The separation of the coefficients into a fac- torial n ! and a reduced coefficient a n often offers formal advantages. Convergence of Factorial Series ⊲ The factorial series for Ω( z ) converges with the possible exception of z = − m with m ∈ N 0 if the associated Dirichlet series ∞ a n ˜ � Ω( z ) = n z n =1 converges. ⇒ The associated Dirichlet series converges as z → ∞ even if a n ∼ n β with β > 0 as n → ∞ . 3

General Considerations ⊲ Stirling apparently became aware about fac- torial series from the work of the French math- ematician Nicole. ⊲ However, Stirling used an thus popularized factorial series in his classic book Methodus Differentialis (1730). ⇒ Later, factorial series played a major role in finite difference equations. Because of = ( − 1) k ( n + k )! n ! ∆ k , k ∈ N 0 , ( z ) n +1 ( z ) n + k +1 it is extremely easy to apply the finite differ- ence operator ∆ (acting on z ) to a factorial series: ∞ a ν ν ! ∆ k Ω( z ) = ∆ k � ( z ) ν +1 ν =0 ∞ a ν ( ν + k )! = ( − 1) k � . ( z ) ν + k +1 ν =0 4

Finite Generating Functions for Stirling Numbers ⊲ Stirling numbers of the first kind: n S (1) ( n, ν ) z ν , � ( z − n + 1) n = n ∈ N 0 . ν =0 ⊲ Stirling numbers of the second kind: n z n = S (2) ( n, ν ) ( z − ν + 1) ν , � n ∈ N 0 . ν =0 Infinite Generating Functions for Stirling Numbers ⊲ Stirling numbers of the first kind: ( − 1) κ S (1) ( k + κ, k ) ∞ 1 � z k +1 = , k ∈ N 0 . ( z ) k + κ +1 κ =0 ⊲ Stirling numbers of the second kind: ( − 1) κ S (2) ( k + κ, k ) ∞ 1 � = , z k + κ +1 ( z ) k +1 κ =0 k ∈ N 0 , | z | > k . 5

Conversion of Inverse Power Series to Factorial Series ⊲ f : C → C possesses a formal inverse power series: ∞ c n � f ( z ) = z n +1 . n =0 ⊲ Inserting the infinite generating function for S (1) ( n, ν ) yields the following factorial series: ∞ m ( − 1) m ( − 1) µ S (1) ( m, µ ) c µ . � � f ( z ) = ( z ) m +1 m =0 µ =0 ⊲ This transformation is purely formal. ⇒ The convergence of the resulting factorial se- ries has to be checked explicitly. ⊲ It can happen that the inverse power series diverges factorially, but the factorial series converges. 6

Stieltjes Functions and Series ⊲ Stieltjes Function � ∞ dΦ( t ) F ( z ) = z + t , | arg( z ) | < π 0 Φ( t ): positive measure on 0 ≤ t < ∞ . ⊲ Stieltjes Series ∞ ( − 1) m µ m /z m +1 � F ( z ) = m =0 � ∞ t n dΦ( t ) µ n = 0 ⊲ We only have to insert the geometric series ∞ ( − t ) ν /z ν +1 = 1 / ( z + t ) � ν =0 into the integral representation and integrate term-wise to obtain the Stieltjes series, which may converge or diverge. ⊲ Stieltjes series are of considerable theoretical importance. There is a highly developed con- vergence theory for their Pad´ e approximants. 7

Waring’s Formula ⊲ Iterating the expression z − w = 1 1 w z + z ( z − w ) yields Waring’s formula ∞ 1 ( w ) n � z − w = Re( z − w ) > 0 , , ( z ) n +1 n =0 which was actually derived by Stirling. ⊲ Inserting the Waring formula with w = − t into the Stieltjes integral yields: � ∞ ∞ ( − t ) n � F ( z ) = dΦ( t ) ( z ) n +1 0 n =0 � ∞ ∞ 1 � = ( − t ) n dΦ( t ) . ( z ) n +1 0 n =0 ⇒ There is a lot of cancellation in the inte- � ∞ gral 0 ( − t ) n dΦ( t ) since ( − t ) n has alternat- ing signs. 8

Factorial Series for Stieltjes Functions ⊲ Inserting the finite generating function for S (1) ( n, ν ) into the Stieltjes integral represen- tation yields: ∞ ( − 1) n � F ( z ) = ( z ) n +1 n =0 � ∞ n t ν dΦ( t ) . S (1) ( n, ν ) � × 0 ν =0 ⊲ Now, we only have to do the moment inte- � ∞ 0 t n dΦ( t ) to obtain a facto- grals via µ n = rial series: ∞ n ( − 1) n S (1) ( n, ν ) µ ν . � � F ( z ) = ( z ) n +1 n =0 ν =0 ⊲ The moments µ n and the Stirling numbers ( − 1) n − ν S (1) ( n, ν ) are always positive . ⇒ There is a lot of cancellation in the strictly alternating finite sum � n ν =0 S (1) ( n, ν ) µ ν rep- resenting the coefficients of the factorial se- ries. 9

Factorial Series for the Euler Integral ⊲ Euler Integral � ∞ e − t d t z + t = e z E 1 ( z ) . E ( z ) = 0 ⊲ Euler Series ∞ ( − 1) n n ! � E ( z ) = z n +1 n =0 Diverges for every finite z but is asymptotic as z → ∞ . ⊲ Factorial Series � ∞ ∞ 1 ( − t ) n e − t d t � E ( z ) = ( z ) n +1 0 n =0 ∞ n ( − 1) n S (1) ( n, ν ) ν ! . � � = ( z ) n +1 n =0 ν =0 10

Summation by Cancellation ν =0 S (1) ( n, ν ) ν ! there is ⊲ In the inner sum � n substantial cancellation: ν =0 S (1) ( n, ν ) ν ! ( − 1) n n ! ( − 1) n � n n 0 1 1 1 -1 -1 2 2 1 3 -6 -2 4 24 4 5 -120 -14 6 720 38 7 -5040 -216 8 40320 600 9 -362880 -6240 10 3628800 9552 11 -39916800 -319296 12 479001600 -519312 13 -6227020800 -28108560 14 87178291200 -176474352 ⇒ The asymptotic Euler series is summed: ( − 1) n � 14 � n ν =0 S (1) ( n, ν ) ν ! n =0 (5) n +1 = 1 . 000 000 764 exp(5) E 1 (5) 11

Conversion of Power Series n =0 γ n z n to an inverse ⊲ Transform f ( z ) = � ∞ power series in 1 /z : ∞ f ( z ) = 1 γ n � (1 /z ) n +1 . z n =0 ⇒ Factorial series in 1 /z : ∞ ( − 1) m f ( z ) = 1 � z (1 /z ) m +1 m =0 m ( − 1) µ S (1) ( m, µ ) γ µ . � × µ =0 ⇒ Use m 1 z z � = (1 /z ) m +1 m ! z + 1 /k k =1 to obtain: ∞ m ( − 1) m z � � f ( z ) = m ! z + 1 /k m =0 k =1 m ( − 1) µ S (1) ( m, µ ) γ µ . � × µ =0 12

Conversion of Factorial Series to Inverse Power Series ⊲ The conversion of factorial series to inverse power series is also possible (although not nearly as useful as the inverse transforma- tion). ⊲ Ω: C → C possesses a factorial series: ∞ w n � Ω( z ) = . ( z ) n +1 n =0 ⇒ Inserting the infinite generating function for S (2) ( n, ν ) yields the following inverse power series: ∞ ( − 1) m � Ω( z ) = z m +1 m =0 m ( − 1) µ S (2) ( m, µ ) w µ . � × µ =0 ⊲ Again, this transformation is purely formal. Convergence has to be checked explicitly. 13

Quartic Anharmonic Oscillator ⊲ Hamiltonian p 2 + ˆ x 2 + β ˆ x 4 . ˆ H ( β ) = ˆ p = − i d ˆ d x . ⊲ Perturbation series (ground state) ∞ b n β n . � E ( β ) = n =0 ⊲ Large-index asymptotics n ! b n ∼ ( − A ) n n 1 / 2 , n → ∞ . ⇒ The perturbation series for E ( β ) diverges fac- torially for every nonzero coupling constant β and has to be summed to produce numer- ically useful results. ⊲ The quartic anharmonic is a simple, but nev- ertheless non-trivial model system for many factorially divergent perturbation expansions in quantum theory. 14

Summation of the Divergent Perurbation Series ⊲ The energy shift ∆ E ( β ) defined by E ( β ) = b 0 + β ∆ E ( β ) ∞ b n +1 β n � = b 0 + β n =0 is known to be a Stieltjes function. ⇒ Pad´ e approximants [ n + j/n ] with j = − 1 , 0 , 1 , . . . to ∆ E ( β ) computed from the divergent per- turbation series converge as n → ∞ . ⊲ Truncated factorial series in 1 /β for ∆ E ( β ): M m ( − 1) m β � � ∆ E ( β ) ≈ m ! β + 1 /k m =0 k =1 m ( − 1) µ S (1) ( m, µ ) b µ +1 . � × µ =0 15