David Vella, Skidmore College dvella@skidmore.edu
Geeratig Fuctios ad Expoetial Geeratig Fuctios Give a sequece {a } we ca associate to it two fuctios determied by power series: Its (ordiary) geeratig fuctio is f x = a x = Its expoetial geeratig fuctio is g x = = a! x
Examples The o.g.f ad the e.g.f of {,,,,...} are: f(x) = + x + x + x 3 + = x, ad g(x) = + x! + x! + x3 3! + = ex, respectively. The secod oe explais the ame... Operatios o the fuctios correspod to maipulatios o the sequece. For example, addig two sequeces correspods to addig the ogf s, while to shift the idex of a sequece, we multiply the ogf by x, or differetiate the egf. Thus, the fuctios provide a coveiet way of studyig the sequeces. Here are a few more famous examples:
eroulli & Euler Numbers The eroulli Numbers are defied by the followig egf: e x x! x The Euler Numbers E are defied by the followig egf: Sech ( x) e e x x 0 E! x
Catala ad ell Numbers The Catala Numbers C are kow to have the ogf: C x = C x = x x = + x = Let S deote the umber of differet ways of partitioig a set with elemets ito oempty subsets. It is called a ell umber. It is kow to have the egf: = S! x = e ex
Higher Order eroulli ad Euler Numbers The th eroulli Number of order w, w is defied for positive iteger w by: e x w w x! x The th Euler Number of order w, E w is similarly defied for positive iteger w as: e x e x + w = = E w! x
Reversig the Process If you start with a fuctio ad ask what sequece geerates it (as a ogf), the aswer is give by Taylor s theorem: c = f (0)! More geerally, if we use powers of (x-a) i place of powers of x, Taylor s theorem gives: c = f (a)! Let us abbreviate the th Taylor coefficiet of f about x = a by: T f; a = f (a)!
A Key Questio I have metioed that operatios o the fuctios correspod to maipulatios of the sequece. What maipulatios correspod to composig the geeratig fuctios? That is, we are askig to express T f g; a i terms of the Taylor coefficiets of f ad of g. I Jauary, 008, I published a paper etitled Explicit Formulas for eroulli ad Euler Numbers, i the electroic joural Itegers. I this paper, I aswer the above questio ad give some applicatios of the aswer.
Sice A Key Aswer T f g; a = f g () (a)!, the aswer would deped o evaluatig the umerator, which meas extedig the chai rule to th derivatives. This was doe (ad published by Faà di ruo i 855.) I oticed (i 99) a corollary of di ruo s formula which exactly aswers the above questio. Fracesco Faà di ruo:
THE MAIN RESULT With the above defiitios, if both y = f(x) ad x = g(t) have derivatives, the so does y = f g(t), ad T f g; a = l π T δ π l π (f; g a ) T i (g; a) πi π P i= where P is the set of partitios of, l π is the legth of the partitio π, ad π i is the multiplicity of i as a part of π. Here, δ π is the set of multiplicities {π i } which is itself a partitio of l π (I call it the derived partitio δ π ), ad multiomial coefficiet l π δ π l π! π!π! π!. is the associated
Illustratio How to use this machie Let f x = e x ad let x = g t = e t Set a = 0 ad observe g a = 0. The: T m f; g 0 = m! for all m ad similarly, T i g; 0 = i! if i. The right side becomes:!!! π P π P δ π! l π δ π! π! =! l(π)! i= i! π i S π = π P S!
Illustratios, cotiued ut the left had side is T f g; 0 for the composite fuctio f g(t) = e et So we just proved this is the egf for the sequece of ell umbers S. This is a very short proof of a kow (ad famous) result. Likewise, I ca provide ew proofs of may combiatorial idetities usig this techique. Ca we discover ew results with this machie? Yes!
Let f x Illustratios, cotiued = l(+x) x ad g t = e t. Agai if a = 0 the g a = g 0 = 0. Also we have f g t = t e t which is precisely the egf of the eroulli umbers. I this case, our machie yields the formulas: ) ( ) ( ) ( ( ) ( P ) m m ( ) m m! S(, m) ad: where S(,m) is the umber of ways of partitioig a set of size ito m oempty subsets (a Stirlig umber of the d kid.)
Illustratios, cotiued This was published i my 008 paper, alog with similar formulas for Euler umbers. These formulas ca be geeralized i at least two ways:. Express the higher order eroulli umbers i terms of the usual eroulli umbers, sice the egf for them is obviously a composite. Similarly for the higher order Euler umbers. This result is still upublished (but I spoke at HRUMC a couple of years ago about it.). Geeralize my corollary of di ruo s formula to the multivariable case. (The ext talk is about this i the case of multivariable eroulli umbers!) For the remaider of this talk, I d like to focus o oe case where the idetity I obtai from my machie is ot obviously useful (or is it?)
Fial Example: the Catala umbers Recall from a earlier slide the geeratig fuctio (ogf) of the Catala umbers: C x = C x = x x = = + x This ca be expressed as a composite geeratig fuctio as follows: Let f u = ad let u = g x = x. The +u C x = f g x. If a = 0 the g a =, so the derivatives ad Taylor coefficiets of f u have to be evaluated at u =. It is easy to check by direct calculatio that f m = ( )m m! m ad therefore T m f; = f m m! = ( )m m.
Catala umbers, cotiued To fid T i (g; 0), we write the radical as a power ad use Newto s biomial series: g x = x = k=0 k x k. k It follows that T i g; 0 = i i. So our formula yields: C = π P l π δ π l(π) l(π) i= i i π i However, we ca simplify this because i= i π i = iπ i =
Simplificatios Thus, C = π P l π δ π +l(π) l(π) i= i π i Next, we rewrite the terms i π i = i + i! π i Whe we take the product of these terms over i, the deomiator becomes i! π i i=, the product of the factorials of all the parts of π, which I abbreviate as π!. Observe that π! =! π Lookig more carefully at the umerator, we obtai:
Simplificatios, cotiued i + π i = 3 i 3 π i We ca factor out a power of i the deomiator, amely i π i = iπ i, ad sice every factor except the first is egative, we ca also factor out a power of -, amely ( ) i πi = iπ i π i. Now whe we take the product over i, this meas we factor out the followig: i the deomiator, iπ i = iπ i = (which cacels part of the outside the sum), ad i the umerator, ( ) iπ i π i = ( ) iπ i π i = ( ) l(π). Combied with the ( ) +l(π) term i frot of the product, this becomes ( ) =. I other words, all the egatives cacel out! Fially, what remais i the square brackets is a product of odd itegers, which we abbreviate with the double factorial otatio (with the covetio that (-)!! = ). The etire thig simplifies to:
New formula for the Catala umbers! We have proved: C =! π P π l π δ π l(π) i 3!! π i i= Let s illustrate this with =. We eed the table: π π π π 3 π l(π) δ(π) [] 0 0 0 [] [3,] 0 0 [ ] [ ] 0 0 0 [] [, ] 0 0 3 [,] 0 0 0 []
Our formula becomes: Catala example ( = ) C =! π P π l π δ π l(π) i 3!! π i i= = 6 + 6 (5!!) + 3 3 3 (!!) + (3!!) + (!!) (( )!!) = 6 5 3 + 3 + 6 + 6 3 8 + 6
Catala example ( = ) = 5 + + + 3 + = This is the correct value as C = 5 8 = Of course it appears as if my formula is kid of useless sice it is so iefficiet! Or maybe ot... The Catala umbers are kow to cout may thigs maybe my formula gives some sort of refiemet of this cout? Speculatio: Dyck words?
The Dyck words of legth 8 AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA AAAA I tried may ways to aturally break these up ito groups of sizes 5,,,3, but always without success. ut the what if we lump the terms together correspodig to partitios of the same legth? This leads to groups of size 5,5,3, ad such groups DO appear aturally i the table... I have some ideas o how to do this i geeral (o proof yet), but I believe that I ca make the idividual terms i my sum always correspod to such groups of Dyck words. Hopefully this will lead to a bijective proof of my formula.
The 008 paper, which has the explicit formulas for the ordiary eroulli & Euler umbers (but ot the higher order oes), ca be dowloaded from this website: http://www.itegers-ejct.org/ Just click o the 008 volume. My paper is the first oe i the Jauary issue.
Appedix: My eroulli umber formula Let s see how it works for = : Legth Derived = Partitios of [] [3,] [,] [,,] [,,,] 3 [] [,] [] [,] [] 30 5 9 3 8 5 3 6 3 3,,, ) (,,, 3 3 3 ) (, ) (,3, ) ( ) (
Appedix: My secod eroulli umber formula For example: m ( ) m m m! S(, m) m! 3 m ( ) m m! S(, m)!7 3!6 3 9 5 30 5!
Appedix: My higher order eroulli umber formula (Vella, Feb., 008 - upublished): w w ( ( )) S i P ( ) w i which expresses the higher order eroulli umbers i terms of the ordiary oes. Here, w (m) is the fallig factorial fuctio w (m) = w w w (w m + ) i
Appedix: Example of Higher Order eroulli formula For example, let s compute. There are 5 partitios of, but oly three of them have legth at most : [], [3,] ad [,]:. 0 6 5 6 6 30 6 3 0 0 0 3 0 [,] () 0 3 0 [,3] () 0 3 0 0 [] () S S S