(1 x n ) 1, (1 + x n ). (1 + g n x n ) r n

Transcription

1 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS HGINGOLD, WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, USA, JOCELYN QUAINTANCE, UNIVERSITY OF PENNSYLVANIA, DEPARTMENT OF COMPUTER SCIENCE, PHILADELPHIA PA 904, USA, Abstract Let f x + a x be a foral power series with coplex coefficiets Let {r } be a sequece of ozero itegers The Iteger Power Product Expasio of f x, deoted ZPPE, is + w x r Iteger Power Product Expasios euerate partitios of ulti-sets The coefficiets {w } theselves possess iterestig algebraic structure This algebraic structure provides a lower boud for the radius of covergece of the ZPPE ad provides a asyptotic boud for the weights associated with the ulti-sets Itroductio I the field of euerative cobiatorics, it iswell ow that + px x, where p is the uber of partitios of [ Equally well ow is the geeratig fuctio for p d, the uber of partitios of with distict parts [ 2 + p d x + x Equatio 2 is a special case of the Geeralized Power Product Expasio, GPPE The GPPE of a foral power series + a x is 3 + a x + g x r, Date: March 0, Matheatics Subect Classificatio 05A7, P8, 4A0, 30E0 Key words ad phrases Power products, geeralized power products, geeralized iverse power products, power series, partitios, copositios, ulti-sets, aalytic fuctios, expasios, covergece, asyptotics

2 2HGINGOLD, WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, where {r } is a set of ozero coplex ubers If r ad g, Equatio 3 becoes Equatio 2 Siilarly, Equatio is a special case of the Geeralized Iverse Power Product Expasio, GIPPE The GIPPE of a foral power series + a x is 4 + a x h x r, where {r } is a set of ozero coplex ubers Equatio is Equatio 4 with r ad h The aalytic ad algebraic properties of the GPPE ad the GIPPE were extesively studied i [5, 6, 4 Sice Equatios ad 2 are geeratig fuctios associated with partitios, it is oly atural to defie a sigle class of product expasios that icorporate both as special exaples Defie the Iteger Power Product Expasio, ZPPE, of the foral power series + a x to be 5 + a x + w x r, wheever {r } is a set of ozero itegers The Equatio 2 is Equatio 5 with r ad w, while Equatio is Equatio 5 with r ad w The purpose of this paper is to study, i a self-cotaied aer, the cobiatorial, algebraic, ad aalytic properties of the ZPPE Sectio 2 discusses, i detail, the role of iteger power product i the field of euerative cobiatorics I particular, we show how iteger power products euerate partitios of ulti-sets We also discuss how the ZPPE factors the foral power series associated with the uber of copositios Sectio 3 derives the algebraic properties of w i ters of {a } ad {r } The ost iportat property, ow as the Structure Property, writes w as a polyoial i {a i } i, whose coefficiets are ratioal expressios of the for pr,r 2,,r We exploit qr,r 2,,r the Structure Property i Sectio 4 whe deteriig a lower boud for the radius of covergece of + w x r Sectio 4 also cotais a asyptotic approxiatio for the iteger power product expasio associated with s x where s sup a, aely the aorizig product expasio 2 Cobiatorial Iterpretatios of Iteger Power Product Expasios Give a foral power series + a x or a aalytic fuctio f x with f 0 which has a Taylor series represetatio + a x, we defie the Iteger Power Product Expasio, deoted ZPPE, as 6 f x + w x r,

3 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 3 where {w } is a set of ozero coplex ubers ad {r } is a set of ozero itegers We say + w x r is a eleetary factor of the ZPPE If r, a eleetary factor has the for + w x r + g x r, while for r, a eleetary factor has the for + w x r h x r If r for all, Equatio 6 becoes the Power Product Expasio f x + g x, while if r for all, Equatio 6 becoes the Iverse Power Product Expasio f x h x Give a fixed set of ozero itegers {r }, there is a oe-to-oe correspodece betwee the set of foral power series ad the set of ZPPE s To discover this correspodece, expad each eleetary factor of Equatio 6 i ters of Newto s Bioial Theore ad the copare the coefficiet of x I particular we fid that + Hece, 7 8 a x 0 r a w x r 2 0 w + l v l < r2 rl v 2 w 2 x r3 3 rlθ w v v l θ w v θ l, θ where l [l, l 2,, l θ ad v [v, v 2,, v θ Equatio 7 iplies that w r a l v l < rl v rlθ w v v l θ w v θ l θ w 3 x 3 3 We foralize the above discussio i the followig propositio which is a stateet about a biectio betwee the sequece of the coefficiets i a give power series ad the sequece of coefficiets i its ZPPE expasio Propositio :Let {r } deote a sequece of ozero itegers Let w C,, 2,, be a ifiite sequece Let the sybol + w x r stad for the ifiite product 9 + w x r : + w x r + w 2 x 2 r2 + w x r The there exists a uique sequece a C,, 2,, such that i the sese of power series the followig holds 0 + a x + w x r Coversely, let a C,, 2,, be a ifiite sequece The there exists a uique sequece of eleets w C,, 2,, such that the idetity 0 holds Moreover, the Olie Joural of Aalytic Cobiatorics, Issue 206, #2

4 4HGINGOLD, WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, eleets w have the represetatio provided by Equatio 8 The oe-to-oe correspodece of Propositio has ay cobiatorial iterpretatios Let be a positive iteger A partitio of is a su of positive itegers i such that i + i i Each i l for l is called a part of the partitio [ Without loss of geerality assue i i 2 i Give i + i i, we associate each part i with the ooial x i The each suad of 0 xi + x i + x 2i + x 3i + represets the part i occurrig ties, ad the product i 0 xi i xi becoes i x i x x 2 x 3 px, 0 where p is the uber of partitios of Equatio is Equatio 6 with r ad w for all To obtai a cobiatorial iterpretatio for Equatio 6 with r ad w we observe that 2 i + x i + x + x 2 + x 3 p d x, 0 where p d couts the partitios of coposed of distict parts [, where a partitio of has distict parts if i + i i ad i l i p if ad oly if l p Equatios ad 2 ay be cobied as follows Let {r } be a set of itegers such that for each, r or r Furtherore require that w r Equatio 6 becoes 3 i + r i x i r i p H x, 0 where p H is the uber of partitios of coposed of uliited uber of copies of the part x if r, ad at ost oe copy of the part x if r For exaple suppose that r i if i is odd ad r i if i is eve Equatio 3 becoes x + x 2 x 3 + x 4 x 5 + x 6 p H x 0 I Equatio 3 we required that r ± Let us reove this restrictio ad ust assue {r } is a arbitrary set of itegers Defie 4, r, sgr, r 0, r 0

5 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 5 Is there a cobiatorial iterpretatio for i + sgr ix i r i? To aswer this questio we eed the otio of a ulti-set Let {r } be a set of oegative itegers Defie the associated ulti-set as r 2 r 2 r, where r deotes r distict copies of the iteger If r 0, there are o copies of i the ulti-set Give {r } 0, a set of positive itegers, we for the geeratig fuctio 5 i + x i r i + x r + x 2 r 2 + x r ˆp d x, 0 where ˆp d couts the partitios of coposed of distict parts of the ulti-set r 2 r 2 i r i To clarify what is eat by distict parts whe worig i the cotext of ulti-sets, it helps to itroduce the otio of color Each of the r i copies of i is assiged a uique color fro a set of r i colors Differetly colored i s are cosidered distict fro each other Thus ˆp d couts the partitios of over the ulti-set r 2 r 2 r which have distict colored parts As a case i poit, tae the ulti-set , ad represet it as { R, B, 2 R, 2 B, 2 O, 2 Y, 3 R, 3 B, 3 O, 4 R, 4 B, 4 O, 4 Y, 4 G } where the color of the digit is deoted by the subscript ad R Red, B Blue, O Orage, Y Yellow, ad G Gree The geeratig fuctio for this ulti-set is 4 + x r + x 2 + x x x 4 5 where expoet of x deotes the part while the expoet of each eleetary factor deotes the uber of colors available for the associated part Equatio 5 is the ulti-set geeralizatio of Equatio 2 There is also a ultiset geeralizatio of Equatio Assue r is a positive iteger Equatio geeralizes as 6 i x i r i x r x 2 r 2 x 3 r3 ˆpx, 0 where ˆp is the uber of partitios of associated with the colored ulti-set which cotais a uliited uber of repetitios of each iteger i r colors I other words, the ulti-set is S r Sr 2 2 Sr i i, where S i {i, i + i, i + i + i, } The factor x i r i + x i + x 2i + x 3i + r i correspods to {i, i + i, i + i + i, } replicated i r i colors As a exaple of Equatio 6, let r 2, r 2 ad r 3 3 The associated geeratig fuctio is x 2 x 2 x 3 3, ad the ulti-set cotais two copies of {, +, + +, }, oe i Red ad oe i Blue; oe copy of {2, 2 + 2, , } i Red; ad three copies of {3, 3 + 3, , } i Red, Blue, ad Orage We cobie Equatios 5 ad 6 as Olie Joural of Aalytic Cobiatorics, Issue 206, #2

6 6HGINGOLD, WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, 7 i + sgr i x i r i ˆp H x, 0 where ˆp H is the uber of partitios coposed fro r i copies of M i, where M i {i, i + i, i + i + i, } if sgr i, ad M i {i} if sgr i As a specific exaple of Equatio 7, let r, r 2 2, ad r 3 2 The M {, +, + +, } occurs i Red, M 2 {2} occurs i Red ad Blue, while M 3 {3, 3 + 3, , } occurs i Red ad Blue, ad the associated geeratig fuctio is x + x 2 2 x 3 2 Equatio 7 is the ulti-set geeralizatio of Equatio 3 To further geeralize Equatio 7 we ultiply each part i of the ulti-set with the weight w i to for 8 i + sgr i w i x i r i ˆp H w, x, 0 where ˆp H w, is a polyoial i {w i } i0 such that each w is the su of ooials w α wα 2 2 wα, where i iα ad α couts the uber of ties colored part appears i the partitio If sgr i, there are r i colored copies of the weighted ulti-set M i {w i i, w i i + w i i, w i i + w i i + w i i } {w i i}, ad each w ii is associated with the ooial wi xi wi xi If sgr i, there are r i colored copies of the weighted ulti-set M i {w i i}, where w i i is associated with the ooial w i x i I the case of the previous exaple with r, r 2 2, ad r 3 2, we ow have oe copy of the weighted ulti-set {w,, w + w, w + w + w, }, two copies of the ulti-set {2w 2 }, ad two copies of the ulti-set, ad the geeratig fuctio is w x + w 2 x 2 2 w 3 x 3 2 The cobiatorial iterpretatios of Equatios through 8 origiated fro the product side of Equatio 6 To develop a cobiatorial iterpretatio fro the su side of Equatio 6, defie f x a x where {a } is a set of positive itegers Equatio 6 iplies that 9 a x + w x r Tae Equatio 9 ad for the reciprocal 20 a x + w x r + w x r

7 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 7 Equatio 20 shows that the reciprocal of a x is also a ZPPE Expad the left side of Equatio 20 as a x + a x C, x + [ C, [ 2 a x + 2 x, [ 3 a x + + C, 2x + + [ a x + C, x + where C, is a polyoial represetatio of the copositios of with exactly parts such that the part i is represeted by a i ad the + is replace by I other words, C, is coposed of ooials ca i a i2 a i such that i + i 2 + i is a partitio of Recall that a copositio of a positive iteger with parts is a su i + i 2 + i where each part i is a positive iteger with i The differece betwee a partitio of with parts ad a copositio of with parts is that a copositio distiguishes betwee the order of the parts i the suatio [?, 2 Our cobiatorial iterpretatio of C, is verified via a stadard iductio arguet o Sice 2 a x + [ C, x : + C x, we ay iterpret C to be the su of all o-trivial polyoial represetatios of the copositios of with parts, ie C is a polyoial represetatio of the copositios of where C is costructed by taig the set of copositios of, replacig i with a i, replacig + with, ad suig the ooials If a, C, is the uber of copositios of with parts, while C is the total uber of copositios of I particular, we fid that 22 x x + x x + x + x + [ 2 x + x 2 x + + x [ 3 x + + x + [ x + Olie Joural of Aalytic Cobiatorics, Issue 206, #2

8 8HGINGOLD, WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, Defie [ x l Ĉl, xl wheever Clearly Ĉl, ad a stadard iductio arguet o shows that Ĉl, l Equatio 22 the becoes + x + x [ 2 x + Ĉ, x + [ Ĉ, 2 x 2 [ 3 x + + Ĉ, 2x + + x + [ [ x + Ĉ, + x Our calculatios have prove of the fact that uber of copositios of is 2, ad the uber of copositios of with parts is See Exaple I6, Page 44 of [2 or Theore 33 of [8 But ore iportatly, by cobiig our observatios with Equatio 20, we see that the ZPPE + w x r provides a way of factorig the series + C x, where C is the polyoial represetatio of the copositios of 3 Algebraic Forulas for coefficiets of Iteger Power Product Expasios I this sectio all calculatios are doe i the cotext of foral power series ad foral power products For a fixed set of ozero itegers {r }, there are three ways to describe the coefficiets of the ZPPE i ters of the coefficiets of a give power series First is Equatio 8 A alterative forula for {w } is foud by coputig the log of Equatio 6 Sice log + w x w x, we observe that 23 log + w x r r log + w x r w x Represet log f x D x Coparig the coefficiet of x s i this expasio of log f x with the coefficiet of x s provided by the expasio i Equatio 23 iplies that 24 D s s Solvig Equatio 24 for w s gives us : s s s r w

9 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 9 25 w s D s s s s r w s s r s Although Equatios 8 ad 25 are useful for explicitly calculatig w, either of these forulas reveal the structure property of w crucial for deteriig a lower boud o the radius of covergece of the ZPPE Tae Equatio 6, defie a C,, ad rewrite it as + C, x + w x r [ + C 2, x, 2 where + 2 C 2,x 2 + w x r Next write + C 2, x + w 2 x 2 r 2 [ + C 3, x, 2 3 where + 3 C 3,x 3 + w x r Cotiue this process iductively to defie 26 + C, x + w x r [ + C +, x, + where + C,x + w x r ad + + C +,x + + w x r By coparig the coefficiet of x o both sides of Equatio 26 we discover that w C, r the followig theore for all This fact, alog with Equatio 26, is the ey to provig Theore 3 Let be ay positive iteger Defie C,0 ad C,N 0 for N Let {r } be a set of ozero itegers Assue that C,N 0 for all N The C +,N 0 wheever + N Proof: Our proof ivolves two cases Case : Assue r The + w x r + g g x r, ad Equatio 26 is equivalet to 27 + [ C +, x + g x r + C, x + Olie Joural of Aalytic Cobiatorics, Issue 206, #2

10 HGINGOLD, 0 WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, Newto s Bioial Theore ad x x+ iplies that + [ C +, x + + [ [ + + [ r g x + C, x [ r + g x + r + C, x [ + r C, x C, x, where the last equality uses the observatio that w g C r If we copare the coefficiet of x s o both sides of the previous equatio we discover that 28 C +,s r + +s r C, C, Equatio 28 ay be rewritte as 29 C +,s A + B, where A : r + +s r 0, C, C, B : r + s s s r C, s r + s 2 s s r C, s We begi by aalyzig the structure of A If r, the r + r + r + 2r r!r is always positive By hypothesis C, 0 ad C, 0 Hece C, C, is either zero or has a sig of + Therefore, r + C, C, is either zero or has a sig of + r

11 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS We ow aalyze the structure of B Uless is a ultiple of s, B vaishes So assue s ˆ where ˆ > The 30 B r +ˆ ˆ r +ˆ 2 C ˆ ˆ rˆ r + ˆ ˆ r +ˆ 2 ˆ ˆ r +ˆ 2 ˆ [ Cˆ, rˆ ˆ r +ˆ 2 ˆ rˆ ˆ r +ˆ 2 ˆ rˆ ˆ Cˆ, rˆ rˆ C, ˆ + ˆ r + ˆ r ˆ + [ r ˆ + + r ˆ Cˆ, r ˆ [ r ˆ Cˆ, r ˆ r +ˆ 2 ˆ Cˆ, rˆ If r the r +ˆ 2 ˆ rˆ ˆ or zero, ad ˆ r +ˆ 2 is positive By hypothesis C, 0 Thus, the sig of Cˆ, is either ˆ rˆ Cˆ, is opositive O the other had, r, with ˆ >, iplies that r ˆ is positive or zero The represetatio of B provided by r ˆ Equatio 30 shows that B is either zero or egative Case 2: Assue r ; that is r is a egative iteger which is represeted as r, ad + w x r h x r Equatio 26 is equivalet to [ + C +, x h x r + C, x + [ [ r + h x + + r C, x r C, x [ + C, x, Olie Joural of Aalytic Cobiatorics, Issue 206, #2

12 HGINGOLD, 2 WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, where the last equality follows fro the fact that w h C If we copare the r coefficiet of x s o both sides of the previous equatio we fid that 3 C +,s Equatio 3 ay be writte as 32 where A : +s 0, +s r C +,s A + B, r r C, C,, B : Sice r is a positive iteger, r r r C,C, r r s s C, s + s s s s C r r, 0 By hypothesis C, C, is the product of + opositive ubers ad is either zero or has a sig of + Thus C, C, is either zero or egative, ad A is opositive It reais to show that B is also opositive Notice that B oly exists if s is a positive iteger, say s ˆ The B becoes r B ˆ ˆ Cˆ, r + r ˆ Cˆ, ˆ ˆ r ˆ ˆ r r Cˆ, ˆ ˆ r + r ˆ ˆ ˆ ˆ [ Cˆ, ˆ r r + r ˆ ˆ ˆ ˆ r ˆ ˆ r ˆ r ˆ r ˆ ˆ r + r ˆ + [ r + ˆ Cˆ, ˆ r ˆ + Cˆ, [ Cˆ, r ˆ Sice r ad ˆ are positive itegers r ˆ 0 By hypothesis Cˆ, is either zero or has a sig of ˆ Thus ˆ r ˆ r ˆ Cˆ, is opositive It reais to aalyze the sig of the ratioal expressio iside the square bracet at 34 The sig of this expressio

13 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 3 depeds oly o the sig of r ˆ sice the other three factors are always oegative If r ˆ + > 0, the r + > ˆ, ad the ratioal expressio is oegative If r + ˆ < 0, the r < ˆ, which i tur iplies that r 0 So oce ˆ agai the quatity at 34 is opositive Oly oe case reais, that of r + ˆ Notice that r ˆ The B ˆ Cˆ, r ˆ, a quatity which is either zero or has a sig of ˆ ˆ I all three cases we have show that B is opositive If we use the otatio of [3, we ay trasfor Theore 3 ito a theore about the structure of the C +,s Defie α, 2,, to be a vector with copoets where each copoet is a positive iteger Let λ λα be the legth of α, ie λ Let α deote the su of the copoets, aely α s s The sybol C,α represets the expressio C, C,2 C, For exaple if α 2, 3, 4, 3, the λ 4, α 2, ad C,2,3,4,3 C,2 C,3 C,4 C,3 C,2 C 2,3 C,4 Theore 32 Structure Property Let be a positive iteger The 35 C +,s λαl cαl,, s C,αl, l where the su is over all uordered sequeces αl, 2, λ such that αl s ad at ost oe i The expressio cαl,, s deotes a ratioal expressio i ters of, s ad r which is oegative wheever r is a positive iteger Furtherore, defie C,αl C, C,2 C,λ If C,s 0 for all oegative itegers ad all s, Equatio 35 is equivalet to 36 C +,s cαl,, s C, C,2 C,λ, where the su is over all uordered sequeces αl, 2, λ such that αl s ad at ost oe i Proof If r, we have Equatio 29 which says C +,s A + B, where A : +s 0, r + r C, C,, B : r + s s s r C, s + s r+ s 2 s s r s C, For A we represet C, C, as C,αl, ad r + as cαl,, s Notice that r λαl For B we cobie via Equatio 30, let Cˆ, C,αl, ad le cαl,, s Olie Joural of Aalytic Cobiatorics, Issue 206, #2

14 HGINGOLD, 4 WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, r +ˆ 2 ˆ rˆ r ˆ r ˆ If r, we have Equatio 32 which says C +,s A + B, where Ā : +s 0, r r C,C,, B : r s r C r s, s + s s s C r s, For A we represet C, C, as C,αl ad r r as cαl,, s Notice that λαl For B we cobie via Equatio 34, let Cˆ, B,αl, ad cαl,, s r ˆ r ˆ r +ˆ ˆ r ˆ+ ˆ r + r r 2 r ˆ+2 r ˆ ˆ! as log as r ˆ + If r ˆ +, the B ˆ Cˆ, ad Cˆ r ˆ, C,αl while cαl,, s r ˆ If we tae Equatio 35 ad iterate ties we discover that 37 C +,s l λαl+ cαl,, s a αl cαl,, s a a 2 a λ, l where where the su is over all αl, 2, λ such that αl s ad cαl,, s is a ratioal expressio i, s, ad { r i } i which is oegative wheever r i is a positive iteger If s + Equatio 37 becoes 38 C +,+ r + w + λαl+ cαl, a αl l cαl, a a 2 a λ, l where the su is over all uordered sequeces αl, 2, λ such that αl + For { r i } + i a set of positive itegers, the coefficiet cαl, is oegative If r +, Equatio 38 iplies that w + g + is egative If r +, Equatio 38

15 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 5 iplies that w + h + is positive We explicitly list w i for i 6 w 0 r a, w 2 r 2r r 2 a r 2 a 2 w 3 2 r2 3r 2r a 3 + a a a 3 3 r 3 r 3 w 4 r 2 2r 2 r 4 a r 2r 2 2r r 2 r 4 a 2 a r 2 + 2r 3 r 2 r 3 + 2r2 r 8r 3 r 2r 4 a 4 + r 4 a a r 4 a 4 w 5 2 r 5 a 2 a 3 + r 5 a 2 a r 5 a a r 5 a 3 a 2 + r 5 a a r4 5r 4r a a 5 5 r 5 w 6 2 r 6 a 2 a 4 + r 6 a 2 a 4 + r 3 2r 3 r 6 a r2 + 3r2 r 3 + 3r 2 r 3r 6 a 3 a r 3 r 3 r 6 a a 2 a r2 2 3r2 2r a r r 3 + 3r r 3 2 2r 3 r r2 2 + r 3 6 2r r2 2r a 2 3r a a 6 6 r r2 2 4r2 r2 2 3r2 r 3 + 6r r 3 + 2r 2 r2 2 r 3 3r 3 2r 2 r2 2 r 3r 6 a 4 a 2 + r 6 a a r5 r2 2 r 3 9r 3 r 3 + 3r 2 r 3 2r 2 2 r 3 3r 5 r 3 + 9r 3 r 4 4r5 r r2 2 r3 4r r r 5 r2 r 3r 6 a 6 4 Covergece Criteria For Iteger Power Products Let {r } be a set of ozero itegers The structure of w provided by Equatio 38 allows us to prove the followig theore Theore 4 Let f x + a x Let {r } be a give set of ozero itegers The f x has ZPPE 39 f x + Cosider the auxiliary fuctios 40 Cx a x + w x r a x sgr Ŵ x r, Olie Joural of Aalytic Cobiatorics, Issue 206, #2

16 HGINGOLD, 6 WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, 4 Mx M x sgr E x r where sgr is defied via Equatio 4 Assue that a M for all The w Ŵ E for all 42 Proof: By Equatio 38 we have w λαl+ cαl, a αl λαl+ cαl, a a 2 a λ, l: αl l: αl Equatio 42 iplies that 43 w l: αl λαl+ cαl, a a 2 a λ cαl, a a 2 a λ l: αl Equatio 38 whe applied to Equatio 40 iplies that 44 0 Ŵ λαl cαl, a a 2 a λ l: αl λ2αl cαl, a a 2 a λ l: αl l: αl cαl, a a 2 a λ Cobiig Equatios 43 ad 44 shows that w Ŵ Sice a M we also have 0 Ŵ cαl, a a 2 a λ cαl, M M 2 M λ E, l: αl l: αl where the last equality follows fro Equatio 38 Thus Ŵ E 45 We ow wor with a particular case of Mx, aely Mx s x sgr E x r, s : sup a We wat to deterie whe the ZPPE of Equatio 45 will absolutely coverget Recall that log sgr E x r r log sgr E x sgr E x r l l l

17 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 7 The 46 log sgr E x r r log sgr E x sgr E x r l l l Equatio 46 iplies that if the double series is absolutely coverget, the both r log sgr E x ad r log sgr E x are absolutely coverget Furtherore, the absolute covergece of the double series iplies the absolute covergece of sgr E x r sice e r log sgr E x e log sgr E x r sgr E x r Thus it suffices to ivestigate the absolute covergece of r log sgr E x If we tae the logarith of Equatio 45 we fid that 47 r log sgr E x log s x Now s x sx 0 sx sx sx 2sx sx Therefore, log 2sx log 2sx log sx sx 2sx + sx 2 sx By the Ratio Test we ow that 2 sx absolutely coverges wheever 2 li + sx < This is esured by requirig x < 2s + 2 We have show that r log sgr E x, ad thus sgr E x r will be absolutely coverget wheever x < 2s We clai this iforatio provides a Olie Joural of Aalytic Cobiatorics, Issue 206, #2

18 HGINGOLD, 8 WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, lower boud o the rage of absolute covergece for the ZPPE of Equatio 39 sice log + w x r r log + w x r log + w x r w x w x r r E x, where the last iequality follows by Theore 4 These calculatios iplies that if r sgr E x, ad hece r log sgr E x, are absolutely coverget, the r log + w x ad + w x r will also be absolutely coverget We suarize our coclusios i the followig theore Theore 42 Let f x + a x Let {r } be a give set of ozero itegers Defie s : sup a The both f x ad its ZPPE, f x + a x ad the auxiliary fuctio, alog with its ZPPE, 48 Mx s x will be absolutely coverget wheever x < 2s + w x r, sgr E x r, We ow provide a asyptotic estiate for the aorizig GIPPE of Equatio 48 Theore 43 Let f x s x 2sx sx where s > 0 Let {r } be a sequece of ozero itegers For this particular f x ad its associated ZPPE + w x r we have 49 r w 2 s, To prove Theore 43 we eed the followig lea Lea 44 Let f x s x 2sx sx where s > 0 Let {r } be a sequece of ozero itegers For this particular f x ad its associated ZPPE + w x r there exists α with < α < 2 such that 50 r w α2 s Proof: A straightforward calculatio shows that r w 69 wheever 2s 30 To prove Equatio 50 for arbitrary assue iductively that r w α2 s is

19 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 9 true for < Our aalysis shows that we ay assue 6 Tae Equatio 24 ad write it as 5 D + r w r w Sice f x s x 2sx sx, log f x log 2sx sx ad we deduce that that D 2s 2 Tae Equatio 5 ad write it as where 2 s x D x, [D + T + T 2 + T 3 + T 4 + T 5 + T 6 + T 7 + r w, T : r w, 7, : r w 2 8 The rage of suatio of iplies that 6 I order to prove Equatio 50 it suffices to show that r w 2s 2s D + T + T 2 + T 3 + T 4 + T 5 + T 6 + T s [ D + T + T 2 + T 3 + T 4 + T 5 + T 6 + T 7 + < 2, wheever 6 We ust approxiate Begi with D 2s ad observe that 2s D, 2s T for 7, ad 2s 53 2s D 2s 2s 2 < Olie Joural of Aalytic Cobiatorics, Issue 206, #2

20 HGINGOLD, 20 WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, We ow wor with T 2s Tae the forulas for w provided at the ed of previous sectio, let a i s i, ad siplify the results to fid that w s r w 2 s2 3r 2r r 2 w 3 s3 7r 2 3r 2 r 3 w 4 s4 9r r3 r 2 + 6r 2 2r 2 r 8r 3 r 2r 4 w 5 s5 3r 4 5r 4 r 5 w 7 s7 27r 6 7r 6 r 7 w 6 s6 4r r 2 2 2r2 2 r 3 + 3r 2 r r 2 2 r3 + 8r 3r r5 r2 2 r 3 96r 5 r2 2 8r5 r 3 27r 3 r 3 72r 5 r2 2 r 3r 6 We use this data to approxiate T 2s for 7 Whe doig the approxiatios recall that r is a ozero iteger for all ad that 6 2s T r s 54 2s r 22 r s T 2 2 r r 2 2r r 2 r2 3r 3r 2 8r r 2 8r r r 2 8 r r r 4r Whe approxiatig T 2s 3 use the fact that T 3 0 if 3 2s T 3 3 r 3 7r r 2r 3 r 3 7r r 2 3 3r 2r 3 r r 2r 3 7r 2 7r r 2r r 2 2s T 4 4 r r 8r r r r3 r 2 + 6r 2 2r 2 r 4 9r 3 8r 3r 4 r4 + 30r3 r 2 + 6r 2 2r 2 r 2r r 3r 2r 4 9r r3 r 2 + 6r 2 2r 2 r r 3r r r3 r 2 + 6r 2 2r 2 r 2 7 r 3 r 2r

21 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 2 Whe approxiatig 2s T 5 use the fact that T 5 0 if s T 5 5 r r 4 5 3r 4 5r 4r 5 r r 4r r 4 3r r r 4r Whe approxiatig 2s T 6 use the fact that T 6 0 if 6 2s T 6 6 r 6 4r r r2 2 r 3 + 3r 2r r2 2r3 + 8r 3r r5 r2 2 r 3 96r 5r2 2 8r5 r 3 27r 3 r r 5r2 2 r 3r 6 4r r2 2 6 r 6 2r2 2 r 3 + 3r 2r r2 2r3 + 8r 3r r5 r2 2 r 3 96r 5r2 2 8r5 r 3 27r 3r r 5r2 2 r 3r 6 4r r2 2 2r2 2 r 3 + 3r 2r r2 2r3 + 8r 3r r5 r2 2 r 3 96r 5r2 2 8r5 r 3 27r 3r r 5r2 2 r 3 4r r2 2 2r2 2 r 3 + 3r 2r r2 2r3 + 8r 3r r5 r2 2 r 3 96r 5r2 2 8r5 r 3 27r 3r r 5r2 2 r 3r Whe approxiatig 2s T 7 use the fact that T 7 0 if 7 2s T 7 7 r r r 6 7r 6r 7 r r 6r r r 6 27r r r 6r Olie Joural of Aalytic Cobiatorics, Issue 206, #2

22 HGINGOLD, 22 WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, 2s It reais to approxiate Here is where we ae use of the iductio hypothesis We also use the fact the 2 iplies 2 By defiitio we have 2s 2s r w α2 2s r s r α 2 8 r α r α 6 α α α r 2 8 α [ α α α 8 8 α 2 8 α α α α [ α 8 α [ 2 α 8 2 α We ow tae Equatios 53 through Equatio 6 ad place the i r w 2s 2s [ D + T + T 2 + T 3 + T 4 + T 5 + T 6 + T 7 + to fid that r w 2s 2s [ D + T + T 2 + T 3 + T 4 + T 5 + T 6 + T < 2 Equatio 52 is valid ad our proof is coplete Proof of Theore 43: Equatio 24 iplies that 62 r w D + r w r w For f x s x we have D 2s 2 Thus Equatio 62 is equivalet to 63 r w 2s 2 + r s + r 2 2 r w Defie

23 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 23 T : 2 + s, Equatio 63 becoes r w T + T 2 + < α < 2 such that s T 2 : r, : r r w 2 2 Lea 44 iplies there exist α with 64 w r w α2 s By defiitio 65 >> α2s 2 2 α2s r w 2 2 r α 2 2 r α α α2s 2 2 r w r 2 2 α2s 2 2 α α2s 2α + 2α 2 + α 3 3 α2s 2α + 2α 2 + 2, 3 3 where the last equality reflects the fact that 2 < α < Defie M : b, : log[ 2 log 2 The 66 b, 2 log r α [ [log 2 + [ α2 s r > 0, 6 Equatio 66 shows that b, is icreasig i wheever 6 Hece b, < b 3, 3 log 2log, 6 6 ad 2 Olie Joural of Aalytic Cobiatorics, Issue 206, #2

24 HGINGOLD, 24 WEST VIRGINIA UNIVERSITY, DEPARTMENT OF MATHEMATICS, MORGANTOWN WV 26506, ad each ter i satisfies eb, e 2log Therefore These calculatios iply that li M 0 By cobiig Equatio 65 with Equatio 67 we deduce that Th li 2 + s li + 2 2s α2s li + 2 2s 2α + 2α M 0 Hece li 0 2 +s We retur to Equatio 63 ad observe that r w T + T s 2 + s 2 + s + r s r + [ r s [ + o D [ + o Rear 45 Theore 43 provides a asyptotic boud for the weights assiged to uderlyig colored ulti-set of Equatio 8 The authors tha the referees for their careful readig of the paper ad thoughtful suggestios regardig iproveet of the expositio Refereces [ George E Adrews, Theory of Partitios, Cabridge Matheatical Library, Cabridge Uiversity Press, Cabridge, UK, 998

25 COMBINATORIAL ANALYSIS OF INTEGER POWER PRODUCT EXPANSIONS 25 [2 P Flaolet ad R Sedgewic, Aalytic Cobiatorics, Cabridge Uiversity Press, Cabridge, UK, 2009 [3 H Gigold, Factorizatio of Matrix Fuctios ad Their Iverses Via Power Product Expasio, Liear Algebra ad its Applicatios, , [4 H Gigold ad A Kopfacher, Aalytic Properties of Power Product Expasios, Ca J Math, 47995, No 6, [5 H Gigold ad J Quaitace, Approxiatios of Aalytic Fuctios via Geeralized Power Product Expasios, Joural of Approxiatio Theory, 88204, 9-38 [6 H Gigold, HW Gould, ad Michael E Mays, Power Product Expasios, Utilitas Matheatica 34988, 43-6 [7 H W Gould, Cobiatorial Idetities, A Stadardized Set of Tables Listig 500 Bioial Coefficiet Suatios, Revised Editio, Published by the author, Morgatow, WV, 972 [8 S Heubach ad T Masour, Cobiatorics ad Copositios ad Words, CRC, Boca Rato, Fl, 2009 Except where otherwise oted, cotet i this article is licesed uder a Creative Coos Attributio 40 Iteratioal licese Olie Joural of Aalytic Cobiatorics, Issue 206, #2