Some polynomials defined by generating functions and differential equations

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1 Dobashi et a, Coget Mathematics & Statistics 208, 4: PURE MATHEMATICS RESEARCH ARTICLE Some poyomias defied by geeratig fuctios ad differetia equatios Nobuyui Dobashi, Eria Suzui ad Shigeru Wataabe * Received: 04 November 206 Accepted: 28 December 206 First Pubished: 06 Jauary 207 *Correspodig author: Shigeru Wataabe, Departmet of Computer Sciece ad Egieerig, The Uiversity of Aizu, Ii-machi Tsuruga, Aizu- Waamatsu City, Fuushima , Japa E-mai: sigeru-w@u-aizuacjp Reviewig editor: Lisha Liu, Qufu Norma Uiversity, Chia Additioa iformatio is avaiabe at the ed of the artice Abstract: It is we ow that geeratig fuctios pay a importat roe i theory of the cassica orthogoa poyomias I this paper, we dea with systems of poyomias defied by geeratig fuctios ad the foowig probems for them A Derive a differetia equatio that each poyomia satisfies B Derive the geera soutio for the differetia equatio obtaied i A C Is the geera soutio obtaied i B writte as a iear combiatio of fuctios that are expressed by maig use of geeraized hypergeometric fuctios? The purpose of this paper is to give two exampes that the probem C ca be affirmativey soved Oe is reated to the Humbert poyomias, ad its geera soutio is writte by F -type hypergeometric fuctios The other is reated to a geearizatio of the Hermite poyomias, ad its geera soutio is writte by F -type hypergeometric fuctios Subjects: Sciece; Mathematics & Statistics; Advaced Mathematics; Aaysis - Mathematics; Specia Fuctios; Differetia Equatios Keywords: Humbert poyomia; geeraized Hermite poyomia; geeraized hypergeometric fuctio; geeratig fuctio; differetia equatio AMS subject cassificatios: Primary: 33C45 Itroductio It is iterestig to defie ew poyomias by ew geeratig fuctios, ad importat to study their properties Humbert 92 defied the poyomias Π ν,m x, 0,, 2,, by the geeratig fuctio mtx + t m ν Π ν,m xt 0 Goud 965 caed Π ν,m x the Humbert poyomia of degree ad gave its geeraizatio Miovaović ad Djordjević 987 gave a differetia equatio for the fuctio Π ν,m x usig differece operators ABOUT THE AUTHORS Eria Suzui ad Nobuyui Dobashi competed their master's theses at the Uiversity of Aizu i 203, 204, respectivey, uder Shigeru Wataabe Some parts of this artice are the mai parts of their master's theses Shigeru Wataabe is a seior associate professor at Departmet of Computer Sciece ad Egieerig, The Uiversity of Aizu, Fuushima, Japa His research iterests are Represetatio Theory of Groups ad Specia Fuctios PUBLIC INTEREST STATEMENT The cassica specia fuctios have various iterestig properties ad appicatios Some geeraizatios for them are aso cosidered For exampe, the sequece of the Humbert poyomias is a geeraizatio of the sequece of the Legedre poyomias I this artice, the authors focus o the Humbert poyomias ad the geeraized Hermite poyomias, ad show that they have deep reatios with the geeraized hypergeometric fuctios 208 The Authors This ope access artice is distributed uder a Creative Commos Attributio CC-BY 40 icese Page of 4

2 Dobashi et a, Coget Mathematics & Statistics 208, 4: Lahiri 97 defied the geeraized Hermite poyomias H,m,ν x, 0,, 2,, by the geeratig fuctio expνtx t m 0 H,m,ν x t! Goud ad Hopper 962 gave the other geeraizatio of the Hermite poyomias by the geeratig fuctio x a x t a exppx r x t r The case of a 0 is equivaet to that defied by Be 934 I Suzui 203, we cosidered defiig the poyomias Q x;, ν, 0,, 2,, by the foowig geeratig fuctio which is simiar to that of the Humbert poyomias 2tx + t ν Q x;, νt, 0 where is a iteger such that 2 ad ν is a positive rea umber Note that Π ν x Q,m mx 2;m, ν ad the poyomia Q x;, ν is ot etirey ew However, we gave a differetia equatio for the fuctio Q x;, ν, which is ot by differece operators ad is a expicit expressio For this reaso, we coud obtai the geera soutio at x 0 of the differetia equatio that Q x;, ν satisfies Ad it is writte as a iear combiatio of fuctios that are expressed by maig use of F -type hypergeometric fuctios I Dobashi 204, we cosidered defiig a geeraizatio of the Hermite poyomias by the geeratig fuctio expt x t +j R x;, jt, 0 where, j are positive itegers Ad we obtaied resuts simiar to the case of Q x;, ν I this case, the correspodig geera soutio is writte as a iear combiatio of fuctios that are expressed by maig use of F +j -type hypergeometric fuctios The purpose of this paper is to give the differetia equatios for Q x;, ν ad R x;, j, ad to derive the geera soutios at x 0 for them The discussio for Q x;, ν is give i Sectio 3, ad that for R x;, j is give i Sectio 4 2 Notatio For a rea umber x, [x] deotes the argest iteger ess tha or equa to x Deote Γα + Γα by α, where Γ is the Gamma fuctio For rea costats a, b, deote axf x+bf x by ax ddx + b f x Deote by N 0 the set of oegative itegers For positive itegers,, meas that is a divisor of The geeraized hypergeometric fuctios F is defied by F α, α 2,, α α ; x m α 2 m α m x m β, β 2,, β β m β 2 m β m m! m0 Page 2 of 4

3 Dobashi et a, Coget Mathematics & Statistics 208, 4: Poyomias Q x;, ν Let be a iteger such that 2, ad ν be a positive rea umber Uess otherwise oted, we fix, ν There exists a positive rea umber δ such that 2tx + t < for x ad δ < t <δ As described i Sectio, we defie the fuctios Q x;, ν, 0,, 2,, by 2tx + t ν Q x;, νt, 0 x, δ < t <δ 2 Lemma The fuctio Q x;, ν has the foowig expressio [ ] Q x;, ν r0 r 2 r ν r+r x r r! r! I particuar, Q x;, ν is a poyomia of degree Proof Sice we have, by the biomia theorem we see that 2tx + t ν 0 0 r0 which impies our assertio ν tt 2x! [ ] 0 r0 ν r! r! 2x r t r r+ r+r ν r+r 2x r t, r! r! 3 Recurrece reatios for Q x;, ν I this subsectio, we sha give recurrece reatios for the fuctios Q x;, ν For x ad δ < t <δ, set φx, t 2tx + t, Φx, t φx, t ν The it is easy to see that the foowig partia differetia equatios hod φx, t t Φx, t+νt 2xΦx, t 0, φx, t Φx, t 2νtΦx, t 0 x We ca derive recurrece reatios for Q x;, ν from these differetia equatios Rewrite both sides of 3 by maig use of 2 The we have 3 4 Page 3 of 4

4 Dobashi et a, Coget Mathematics & Statistics 208, 4: Q + x;, νt 2xQ x;, νt Q + x;, νt 2νxQ x;, νt νq + x;, νt 0 Compare the coefficiets of t i both sides of 5 The we have + ν + Q + x;, ν+ + Q + x;, ν 2 + νxq x;, ν 0, Simiary, by 4, we have the foowig recurrece reatio Q x;, ν 2νQ x;, + ν+2xq x;, ν Q + x;, ν, 7 Further, we sha give some recurrece reatios for the fuctios Q x;, ν that ca be derived from the Equatios 6 ad 7 Differetiate both sides of 6 ad substitute 7 ito it The we obtai 2ν + νq x;, ν+2ν xq x;, ν ν Q + x;, ν 0, Remar we have This recurrece reatio hods aso for 0 < I fact, if 0 <, from Lemma Q x;, ν 2 ν x, Q! + x;, ν 2+ ν + +! x+, which assert that the recurrece reatio above hods aso for 0 < That is, we have 2ν + νq x;, ν+2ν xq x;, ν ν Q + x;, ν 0, 0 8 Sove the Equatio 7 for Q + x;, ν ad substitute it ito 8 The we have ν Q x;, ν 2ν Q + x;, ν + 2ν xq x;, ν, 9 32 Differetia equatio that Q x;, ν satisfies I this subsectio, by maig use of the resuts give i the precedig subsectio we sha give a differetia equatio that Q x;, ν satisfies The mai theorem is Theorem For a arbitrary 0, the fuctio Q x;, ν satisfies the foowig differetia equatio Q 2 x;, ν { x ddx } + + ν + r xq x;, ν Q x;, ν Page 4 of 4

5 Dobashi et a, Coget Mathematics & Statistics 208, 4: Proof Operate d m dx m to both sides of 8 ad mae use of the Leibiz rue The we have ν Q m+ + Suppose that ν This equatio is equivaet to Repacig m by m ad by, the we have Repeatig this formua, the we obtai Set m The, we have x;, ν 2ν xqm+ x;, ν + 2ν + ν + m Q m x;, ν, m, 0 Q m+ x;, ν 2 { x ddx } + + m + ν + m Q m x;, ν, m, 0 Q m x;, ν 2 { x ddx } + m + ν + m Q m x;, ν, m, Q m x;, ν 2 { x ddx } + m + ν + m 2 { x ddx } + m + ν + m 2 2 { x ddx } + m + ν + m Q m x;, ν, Q x;, ν 2 m, { x ddx } + + ν + r Q + x;, ν, Maig use of 9, it is easy to see that our assertio hods for That is, for the foowig hods Q 2 x;, ν { x ddx } + + ν + r xq x;, ν Q x;, ν 0 I the case of 0 <, cosiderig Remar, we see that xq x;, ν Q x;, ν 0, which impies that 0 hods aso for 0 < Therefore, if ν, we ca cocude that 0 hods for ay 0 I the case of ν, ote that both sides of 0 are cotiuous with respect to ν Taig the imit ν i 0, we ca see that 0 hods aso for ν ad 0 Exampe If 2, the poyomia Q x;, ν is equa to the Gegebauer poyomia of degree ad the differetia equatio i Theorem is as foows x 2 Q x;2, ν 2ν + xq x;2, ν+ + 2νQ x;2, ν 0, which is we ow as Gegebauer s differetia equatio Page 5 of 4

6 Dobashi et a, Coget Mathematics & Statistics 208, 4: Geera soutio of differetia equatio that Q x;, ν satisfies I this subsectio, we sha give the geera soutio at x 0 of the differetia equatio that Q x;, ν satisfies By Theorem, the differetia equatio that we cosider is as foows y { x ddx } ν + r xy y To sove this equatio, we use the power series method Sice x 0 is a reguar poit of the Equatio, we set y a m x m m0 2 It is cear that y ma m x m, y m m0 m +! a m! m+ x m 3 Substitute 2, 3 ito Hece, we obtai m +! a 2 m! m+ x m m0 { { } } m a m m + + ν + r x m m0 4 Comparig the coefficiets of x m i 4, the we have m +! a 2 m! m+ { { } } m a m m + + ν + r, which is equivaet to a 2 m { } m!m m + + ν + r m! a m 5 For each N 0 ad each p N 0 such that 0 p <, set m + p Repeatig 5, it is easy to see that +p! 2+p! p! +p 2+p p a +p + p! +p! + p! 2 a p where A r, 6 Page 6 of 4

7 Dobashi et a, Coget Mathematics & Statistics 208, 4: A r +p + + ν + r 2+p + + ν + r p + + ν + r By simpe cacuatios, we have +p 2+p p + p 2 + p p p, ad A r p + + ν + r p + + ν + r p + + ν + r p + + ν + r 7 8 Further, we ow the foowig formua cf Prudiov, Brychov, & Marichev, 986, p + p p! + p + p!! 9 Substitute 7, 8 ad 9 ito 6 Hece, we obtai 2 p p + +ν+ r a +p a p p+ p+2 + p+! Thus, we have a +p x +p a p x p 0 F p Therefore, we obtai the desired resut That is, set F,ν,p x F p We obtai the foowig, p + +ν+,, Theorem 2 The fuctios x p F,ν,p x p 0,,, are the ieary idepedet soutios at x 0 of I particuar, its geera soutio at x 0 is give by p + +ν+ p+, p+2,,, +,, p+, p + +ν+,, p + +ν+ p+, p+2,,, +,, p+ ; 2x ; 2x Page 7 of 4

8 Dobashi et a, Coget Mathematics & Statistics 208, 4: a p x p F,ν,p x, p0 where a 0,, a are arbitrary costats Remar 2 We choose, p N 0 such that + p, 0 p < The F,ν,p x is a poyomia of degree Thus, x p F,ν,p x is a poyomia of degree, ad differs oy by a costat from Q x;, ν 4 Poyomias R x;, j Let, j be positive itegers Uess otherwise oted, we fix, j As described i Sectio, we defie the fuctios R x;, j, 0,, 2,, by expt x t +j For N 0, set I, j, { R x;, jt, < x <, < t < 0 q N 0 0 q [ ] }, + jq + j 20 2 The we obtai Lemma 2 The fuctio R x;, j has the foowig expressio R x;, j q I,j, q x +jq! q! +jq If I, j,, we mea R x;, j 0 Proof Maig use of the Tayor expasio for the expoetia fuctio exp x, it is easy to see that expt x t +j expt x exp t +j t x p t +j q p! q! p0 q0 q x p t p++jq p! q! p,q0 For N 0 there exist p, q N 0 such that p + + jq if ad oy if p + jq, which are equivaet to + jq, [ ] 0 q, + j [ ] 0 q + j Therefore, we obtai the desired resut Page 8 of 4

9 Dobashi et a, Coget Mathematics & Statistics 208, 4: Recurrece reatios for R x;, j I this subsectio, we sha give recurrece reatios for the fuctios R x;, j Set Φx, t expt x t +j The it is easy to see that the foowig partia differetia equatios hod t Φx, t t x + jt +j Φx, t, x Φx, t t Φx, t We ca derive recurrece reatios for R x;, j from these differetia equatios Rewrite both sides of 22 by maig use of 20 The we have + R + x;, jt xr + x;, jt +j + jr j+ x;, jt Compare the coefficiets of t i both sides of 24 The we have + R + x;, j xr + x;, j + jr j+ x;, j, + j Simiary, by 23, we have the foowig recurrece reatio R x;, j R x;, j, 26 Repacig by j +, the we have R x;, j R j+ j+ x;, j, + j 27 Substitute 26, 27 ito 25 We have + R + x;, j xr + x;, j + jr j+ x;, j, + j Repacig by, the we obtai + jr x;, j j xr x;, j R x;, j, + j Remar 3 This recurrece reatio hods aso for + j > j Suppose that + j > j The it is easy to see that [ ] 0, + j It foows from these reatios ad Lemma 2 that R x;, j [ ] j 0 + j { x!,, 0, otherwise, Page 9 of 4

10 Dobashi et a, Coget Mathematics & Statistics 208, 4: ad R j x;, j Thus, we obtai {, j 0, 0, otherwise xr x;, j R x;, j 0, R j x;, j 0 Therefore, we ca cocude that + jr x;, j j xr x;, j R x;, j, j 42 Differetia equatio that R x;, j satisfies 28 I this subsectio, by maig use of the resuts give i the precedig subsectio we sha give a differetia equatio that R x;, j satisfies The mai theorem is Theorem 3 For a arbitrary 0, the fuctio R x;, j satisfies the foowig differetia equatio R +j x;, j + j {x ddx + r + j } R x;, j Proof Operate d m dx m to both sides of 28 ad mae use of the Leibiz rue The we have + jr m+ j x;, j xr m+ m 0, j x;, j+m R m x;, j, Repacig m by m ad by j, the we have R m x;, j {x ddx } j + j + m j R m x;, j, j m, j Repeatig this formua, the we obtai R m x;, j {x ddx } j + j + m j {x ddx } + j + m 2 j 2 {x ddx } + j + m j R m x;, j, m, j Set m The we have R j x;, j + j j {x ddx + r j r } R x;, j, 29 O the other had, by 26 we see that Page 0 of 4

11 Dobashi et a, Coget Mathematics & Statistics 208, 4: R j x;, j R j x;, j, j 30 It foows from 29, 30 that R +j x;, j + j j {x ddx + r + j } R x;, j, 3 I what foows, we assume that 0 < j I this case, we have + j < j + j < Tae q I, j, It foows from 2, 32 that 0 q < ad there exists r N 0 such that 32 0 < r, r q For such r, we see that {x ddx } q + r + j x +jq 0 +jq! q! Thus, by Lemma 2 we obtai {x ddx + r + j } R x;, j 0 33 Next, we sha show R +j x;, j 0 34 Set +,, N 0, 0 <, where ote that < j By this reatio, 26 ad Lemma 2, we have R x;, j R x;, j R x;, j q I,j, q x +jq! q! +jq 35 Notice that R x;, j {, 0, 0, otherwise It foows from this reatio ad 35 that 34 Therefore, by 33, 34 we ca cocude that our assertio hods aso for 0 < j Exampe 2 If j, the poyomia R x;, j is essetiay equa to the Hermite poyomia of degree ad the differetia equatio i Theorem 3 is as foows R x;, 2 xr x;, + 2 R x;, 0, which is we ow as Hermite s differetia equatio Page of 4

12 Dobashi et a, Coget Mathematics & Statistics 208, 4: Geera soutio of differetia equatio that R x;, j satisfies I this subsectio, we sha give the geera soutio at x 0 of the differetia equatio that R x;, j satisfies By Theorem 3, the differetia equatio that we cosider is as foows y +j + j {x ddx + r + j } y 36 To sove this equatio, we use the power series method Sice x 0 is a reguar poit of the Equatio 36, we set y a m x m m0 37 Substitute 37 ito 36 Hece, we obtai m0 m + + j! a m! m++j x m { } a + j m{ } m + r + j x m m0 38 Comparig the coefficiets of x m i 38, the we have m + + j! { } a m! m++j a + j m m + r + j, which is equivaet to a m m j! + j m r jr m! a m j 39 For each N 0 ad each p N 0 such that 0 p < + j, set m + j + p Repeatig 39, it is easy to see that + j +p! + j 2+p! p! a +j+p + j + p! + j +p! + j + p! a p B + j r, where 40 B r + j +p r jr + j 2+p r jr p r jr By simpe cacuatios, we have Page 2 of 4

13 Dobashi et a, Coget Mathematics & Statistics 208, 4: B r + j p r jr + j p r jr + j p r jr + j p r jr + j + j 4 Further, repace by + j i 9 ad substitute it ad 4 ito 40 Hece, we obtai a +j+p + j +j p+ +j Thus, we have a +j+p x +j+p a p x p 0 a p p+2 F +j p r jr +j +j +j+ Therefore, we obtai the desired resut That is, set +j p j +j p+, p+2 +j +j, +j p 2 2j +j,, +j, +j +j,, p++j +j+ +j +j! p 2 j +j,, p++j +j +j ; x + j G,j,p x F +j p j +j p+, p+2 +j, p 2 2j +j,, +j, +j +j,, +j+ +j p 2 j,, +j p++j +j +j ; x + j We obtai the foowig Theorem 4 The fuctios x p G,j,p xp 0,,, + j are the ieary idepedet soutios at x 0 of 36 I particuar, its geera soutio at x 0 is give by +j p0 a p x p G,j,p x, where a 0,, a +j are arbitrary costats Remar 4 Assume that R x;, j is ot ideticay equa to 0 The we have I, j, Tae q I, j, ad set + jq p, p N 0, p + jq + p, q, p N 0,0 p < + j, q q + p, q, p N 0,0 p <, p p, r p The we have p r jr + j q q, which meas G,j,p x is a poyomia Thus, x p G,j,p x is a poyomia soutio of 36, ad differs oy by a costat from R x;, j Page 3 of 4

14 Dobashi et a, Coget Mathematics & Statistics 208, 4: Fudig The authors received o direct fudig for this research Author detais Nobuyui Dobashi E-mai: otetu32@gmaicom Eria Suzui E-mai: seasideaieto@gmaicom Shigeru Wataabe E-mai: sigeru-w@u-aizuacjp Departmet of Computer Sciece ad Egieerig, The Uiversity of Aizu, Ii-machi Tsuruga, Aizu-Waamatsu, Fuushima , Japa Citatio iformatio Cite this artice as: Some poyomias defied by geeratig fuctios ad differetia equatios, Nobuyui Dobashi, Eria Suzui & Shigeru Wataabe, Coget Mathematics & Statistics 208, 4: Refereces Be, E T 934 Expoetia poyomias Aas of Mathematics, 35, Dobashi, N 204 Some ew fuctios associated with cassica specia fuctios Master s thesis, Aizu- Waamatsu: The Uiversity of Aizu Goud, H W 965 Iverse series reatios ad other expasios ivovig Humbert poyomias Due Mathematica Joura, 32, Goud, H W, & Hopper, A T 962 Operatioa formuas coected with two geeraizatios of Hermite poyomias Due Mathematica Joura, 29, 5 63 Humbert, P 92 Some extesios of Pichere s poyomias Proceedigs of the Ediburgh Mathematica Society, 39, 2 24 Lahiri, M 97 O a geeraisatio of Hermite poyomias Proceedigs of the America Mathematica Society, 27, 7 2 Miovaović, G V, & Djordjević, G 987 O some properties of Humbert s poyomias Fiboacci Quartery, 25, Prudiov, A P, Brychov, Y A, & Marichev, O I 986 Itegras ad Series New Yor, NY: Gordo & Breach Suzui, E 203 New properties of dis poyomias ad some fuctios associated with cassica specia fuctios Master s thesis Aizu-Waamatsu: The Uiversity of Aizu 208 The Authors This ope access artice is distributed uder a Creative Commos Attributio CC-BY 40 icese You are free to: Share copy ad redistribute the materia i ay medium or format Adapt remix, trasform, ad buid upo the materia for ay purpose, eve commerciay The icesor caot revoe these freedoms as og as you foow the icese terms Uder the foowig terms: Attributio You must give appropriate credit, provide a i to the icese, ad idicate if chages were made You may do so i ay reasoabe maer, but ot i ay way that suggests the icesor edorses you or your use No additioa restrictios You may ot appy ega terms or techoogica measures that egay restrict others from doig aythig the icese permits Coget Mathematics & Statistics ISSN: is pubished by Coget OA, part of Tayor & Fracis Group Pubishig with Coget OA esures: Immediate, uiversa access to your artice o pubicatio High visibiity ad discoverabiity via the Coget OA website as we as Tayor & Fracis Oie Dowoad ad citatio statistics for your artice Rapid oie pubicatio Iput from, ad diaog with, expert editors ad editoria boards Retetio of fu copyright of your artice Guarateed egacy preservatio of your artice Discouts ad waivers for authors i deveopig regios Submit your mauscript to a Coget OA joura at wwwcogetoacom Page 4 of 4