The Iverse Melli Trasform, Bell Polyomials, a Geeralized Dobiski Relatio, ad the Cofluet Hypergeometric Fuctios Tom Copelad Tsukuba, Japa tcjp@ms.com Nov. 6, 2 Iverse Trasform Represetatios of Delta Fct.. Iverse Fourier Trasform δy sil y lim lim l y l l/2 l/2 e x y e x y The limits, as usual, are to be take outside ay itegral cotaiig the delta fct..2 Iverse Laplace Trasform Lettig p x i the iverse Fourier Traform gives δy i i e p y dp..3 Iverse Melli Trasform i I δz w i e p z w dp, let z ly, w lx, ad p s with δ[ly lx] i i y s x s ds. But, x, y >, givig
INVERSE TRANSFORM REPRESENTATIONS OF DELTA FCT. 2 δ[fu] u i zeros of f δ[ dfu i du u u i ] zeros of f δ[ u u i ] / dfu i, so du δ[ly lx] δ[lx ly] y δ[x y]. Therefore, δy x δx y i i y s x s ds The formally, with Hx the Heaviside step fuctio, Hx fx fy δy x dy σ+ i σ i, the iverse Melli Trasform rep. fy y s dy x s ds σ+ i σ i fs x s ds, i.e., the iverse Melli Trasform of fs, idetifyig the Melli trasform of fx as fs fx x s. A appropriate cotour must be chose to obtai the desired fx usig the iverse Melli trasform. Example I: Widow Fuctio Hy H y δy x Hy { σ+ i σ i y s x s ds} Hy σ+ i σ i y s { x s } ds Hy σ+ i σ i y s { } ds for σ < s Hy σ+ i σ i y s { } ds for σ > s where Hx is the Heaviside step fuctio that vaishes for x <, equals 2 ad is for x >. for x,
2 INTERPOLATION AND RAMANUJAN S MASTER FORMULA 3 Coversely, fs H x x s x s s for Res>. Example II: Dirichlet Series Let fx the fs a a δ x a δ x x s δ x x s a s, a Dirichlet series. Coversely, fx a σ+ i σ i fs x s ds σ+ i σ i / s x s ds a σ+ i σ i a δ x s x s ds a δ x. 2 Iterpolatio ad Ramauja s Master Formula With the fiite differece operator s a s a, s j xj xs xj, the Newto iterpolatio of for Res >. j! s! j! Defiig the modifed Melli Trasform ad its iverse as fs Hx fx the formally fx xs s! ad σ+i s! σ i fs x s ds σ+i sis σ i fs x s s! ds,
2 INTERPOLATION AND RAMANUJAN S MASTER FORMULA 4 fs fx xs s! fx s j xj j! s j Hx fx Hx fx xj s j! j f x! <RMF> fj +, a Newto iterpolatio, ad by closig the iversio cotour to the left with Reσ > ad pickig up the sigularities of if fx is such that fs has o sigularities ad decays sis sufficietly fast i the eighborhood of ifiity. Example III: Expoetial Fct. fs Fiite Part {e x [ e x Hx {e x [ j e x xs s! x j ]} xs for <Res< j! s! xs for Res>. s! j x j j! ]} σ+i σ i s! x s ds σ+i x s ds sis s! σ i for < σ <. Hx e x for σ >. σ+i σ i s! x s ds σ+i x s ds sis s! σ i fs s j fj + s j for all s.
3 BELL / EXPONENTIAL / TOUCHARD POLYNOMIALS 5 3 Bell / Expoetial / Touchard Polyomials For the umber operator x d, x s s! is a eigefuctio with eigevalue s, i.e., x d x s s! s x s s!. For σ > ad, a atural umber, Hx e x [x d ] exp x Hx e x [x d ] σ+i x s ds sis s! σ i Hx e x σ+i sis s x s ds s! σ i Hx e x j j j xj j! Hx j j k k xj j! Hxφ x Hx e x φ : x d : e x, where φ x is the th Bell polyomial, : x d :k x k dk k. The biomial trasform has bee used above: e x e a. x e a. x or e x j j a j x j j! j j k a k xj j!. Coversely, MMT [e x φ x] e x φ x xs s! s for Res> s k k j [ j + ] for all s, agreeig with the Newto iterpolator fs s k k j fj +. To determie the expoetial geeratig fct. for the polyomials cosider Hx exp[t φ. x] Hx e x exp[t φ. : x d :] e x Hx e x expt x d e x e x expt x d σ+i x s ds sis s! σ i
Hx e x Hx e x [ j σ+i sis σ i x s exp t s ds for Res > s! j e t j xj j! ] Hx exp[ x et ] Hx e x exp[ e t : x d :] e x. So also expt x d Gx exp[t φ. : x d :] Gx exp[ et : x d Gx e t. :] Gx 4 Geeralized Dobiski Relatios: Cofluet Hypergeometric Fuctios / Geeralized Laguerre Fuctios The results of the previous sectios ca formally be geeralized operatioally as Hx e x fx d e x Hx e x Hx e x fx d σ+i sis σ i f s x s s! ds σ+i x s ds sis s! σ i Hx e x f[ φ. : x d Hx f[ φ. x], :] e x givig a geeralized Dobiski relatio, Hx f[ φ. x] Hx e x σ+i sis σ i f s x s s! ds, with, for Res > σ, the Melli trasform givig the umbral hybrid Laplace-Melli trasform f s e x f[ φ. x] x s s!.
Checkig for cosistecy, substitute φ. y for s, the formally Hy f φ. y e x f[ φ. x] e y e x f[ φ. x] e y e x f[ φ. x] e y e x f[ φ. x] e y Hy f[ φ. y], ad also e x f[ φ. x] : x! x φ. y [ φ. y ]! : y! δ x y! exp y d δx δx y : x φ. y [ φ. y ]! : Hy ey σ+i sis σ i x s y s ds s! s! Hy e y Lettig fx f φ. x σ+i σ i x+α+ φ. x+α+ x y x s ds Hy e y δx y., the α+ K, α +, x L α x where Ka, b, x is Kummer s cofluet hypergeometric fuctio ad L α x, a geeralized Laguerre fuctio. The Hx L α x Hx φ. x+α+ Hxe x Hx j Hx e x j+α+ σ+i sis σ i x! Hx s+α+ +α+ α+ x α+! x s s! ds. x!
For the associated Newto iterpolatio cosider the Melli trasform agai. f s j e x f[ φ. x] x s s!, e x e x e x fj j f x! fj x! e x x! x s s! x s s! x s s! j fj s + s j fj s j fj. Summarizig, the associated Newto iterpolatio is f s s j fj, or chagig variables, fs s j fj, or traslatig the fuctio, fs s j fj +. x+α+ Specifically for fx Chu-Vadermode idetity appears: s+α+ e x e x φ. x+α+ α+ x α+!, we walk through the steps agai to see that the x s s! x s s!
s j α+ +s α+ α+ s j+α+. x+α+ s j+α+ j Substitutig s for s ad α + for α or traslatig the fct. gives s+α+ s j++α+ j, or for fx, fs s j fj +. Also, s+α+ e x e x e x φ. x+α+ +α+ x! +α+ x! x s s! x s s! x s s!, ad cosistetly j+α+ j dj j e x φ. x+α+ +α+ x! x j j! x j j! +α+ x δ j x! e x φ. x+α+ e x φ. x+α+ δ j x x
Now apply the umbral compositioal iverse of φ x to these equatios. For fx x, from the geeralized Dobiski relatio, Hx f φ. x Hx φ x Hx e x Hxe x j j j xj Hx k j! j k j xk k!. σ+i sis σ i s x s s! ds Replace x umbrally by the the egative of the fallig factorial x., the φ [x. ] k j k j k x k k! x j j x. The last equality follows from the Melli trasform results above with fx x : f s e y f[ φ. y] y s s! dy, givig with x s, x e y e y φ y y x x! k j k j yk k! dy y x x! dy x j j. So, the fallig factorial is the umbral compostioal iverse of the Bell polyomial. Recallig the formulas from above Hx L α x Hx φ. x+α+ Hxe x Hx j Hx e x j+α+ σ+i sis σ i x! Hx s+α+ +α+ α+ x α+! x s s! ds, x! ad substitutig the egative of the fallig factorial s. for x gives
s+α+ s j j+α+ s α+ α+ ad from e x L α x +α+ s σ+i siω σ i ω+α+ s ω ω dω?diverges? s si siα+ k α+ k α+k s k k α+ α+k s +. α+ Note: +α+ s si siα+ s α+ ca be obtaied by repeated use of siλ λ λ! λ! or by usig the Kummer trasformatio pg. 55, Abramowitz ad Stegu, Hadbook of Mathematical Fuctios to obtai e x L α x 4.. Exercises si siα+ Lα α++ x. A Cofirm that e t t φ. x+α φ. x+α! t! dt φ. x+α+ L α x. B Show that t φ. x φ. x! et δx t e x δx t. C Show that φ. x + x, Re >, x >.! D Show that where Uα,, x φ. x α E Show that ψα,, x ψ, α +, x L α x. L α x U,α+,x! is the Tricomi cofluet hypergeometric fuctio. φ. x Uα,,x α α!, so that
F Usig show that siλ λ ν ν µ+ µ+λ siα α Reν >, L α x xα α >, x >. α! G Iterpret! D t t φ. x++α t where D t tλ λ! tλ λ! F P! [: D tt :] t φ. x+α t FP z t t t t z! z λ λ! dz FPFiite Part! z t + z λ λ! dz ad [: D t t :] D t t. H Iterpret α! W α t t φ. x+ α t where W α t α! [: W tt :] α t φ. x+ t F P I Iterpret α! D α t t φ. x t J Prove the relatio t z t α α! dz ad [: W t t :] α W α t t α. α! [: D tt :] α t φ. x +α t. L α x + y e y k L α+k x yk L α+φ. y k! x k φ. x y+α+ φ. x + φ. y+α+ startig with s j s+α+ K Explore! φ. x + j+α+.! L φ. x++!l x. x ad! Which of the above expressios are more fudametal,.i.e., are valid as they stad, for these two cases?