Explicit Formulas and Combinatorial Identities for Generalized Stirling Numbers

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1 Explicit Formulas ad Combiatorial Idetities for Geeralized Stirlig Numbers Nead P Caić, Beih S El-Desouy ad Gradimir V Milovaović Abstract I this paper, a modified approach to the multiparameter o-cetral Stirlig umbers via differetial operators, itroduced by El-Desouy, ad ew explicit formulae of both ids of these umbers are give Also, some relatios betwee these umbers ad the geeralized Hermite ad Truesdel polyomials are obtaied Moreover, we ivestigate some ew results for the geeralized Stirlig-type pair of Hsu ad Shiue Furthermore some iterestig special cases, ew combiatorial idetities ad a matrix represetatio are deduced Mathematics Subject Classificatio 2010 Primary 11B73; Secodary 05A10, 05A19 Keywords Multiparameter o-cetral Stirlig umbers, geeralized Stirligtype umbers, differetial operators, geeratig fuctio, combiatorial idetities 1 Itroductio ad prelimiaries Through this article we use the followig otatios The fallig ad risig factorials are defied, respectively by ad x = xx 1x + 1, x 0 = 1, x = xx + 1x + 1, x 0 = 1 The geeralized fallig ad risig factorials x;ᾱ ad x;ᾱ, associated with parameter ᾱ = α 0,α 1,,α 1 where α j, j = 0,1,, 1, is a sequece This paper was supported i part by the Serbia Miistry of Educatio ad Sciece Caić ad Milovaović ad by the Egyptia Miistry of Higher Educatio El-Desouy Correspodig author

2 2 N P Caić, B S El-Desouy ad G V Milovaović of real or complex umbers, are defied by 1 x;ᾱ = x α j ad x;ᾱ = 1 x + α j, respectively Note that if α i = iα, i = 0,1,, 1, the x;ᾱ reduces to x α = tt αx 1α Hsu ad Shiue [14] defied geeralized Stirlig-type pair by the iverse relatios ad {S 1,,S 2,} {S,;α,β,r,S,;β,α, r} x α = x β = S 1,x r β 11 S 2,x + r α, 12 where is a oegative iteger ad the parameters α,β ad r are real or complex umbers, with α,β,r 0,0,0 For the umbers S,;α,β,r, we have the followig recurrece relatio S + 1,;α,β,r = S, 1;α,β,r + β α + rs,;α,β,r, 13 where 1 These umbers have the vertical geeratig fuctios with αβ 0! S,;α,β,r x 1 + αx β/α 1 = 1 + αxr/α 14 0! β ad! S,;β,α, r x 1 + βx α/β 1 = 1 + βx r/β 0! α The multiparameter o-cetral Stirlig umbers of the first ad secod id, respectively, were itroduced by El-Desouy [11] with ad ad x = s,;ᾱx;ᾱ 15 x;ᾱ = S,;ᾱx 16 These umbers s,;ᾱ ad S,;ᾱ satisfy the recurrece relatios respectively s + 1,;ᾱ = s, 1;ᾱ + α s,;ᾱ 17 S + 1,;ᾱ = S, 1;ᾱ + α S,;ᾱ, 18

3 Explicit Formulas ad Idetities for Geeralized Stirlig Numbers 3 The multiparameter o-cetral Stirlig umbers of the first id have the vertical expoetial geeratig fuctio s,;ᾱ x 0! = 1 + x α j α j, 19 where α l = l, j α α j, l Also, more geeralizatios ad extesios of Stirlig umbers are give i [4] ad [20] [22] Let δ = xd, D = d/dx The we have some well-ow ad useful relatios of this differetiatio operator see [2, 10, 16] ad [19]: i δ x α = α x α, ii δ u v = δ u δ v, iii iv Fδx α f x = x α Fδ + α f x, Fδe gx f x = e gx Fδ + xg x f x The paper is orgaized as follows I Sectio 2 a modified approach via differetial operator to multiparameter o-cetral Stirlig umbers is give Also, some relatios betwee these umbers ad the geeralized Hermite ad Truesdel polyomials are obtaied Moreover, we show that some of Hsu ad Shiue [14] results are ivestigated usig the results obtaied by El-Desouy [11] ad cosequetly of this paper I Sectio 3, ew explicit formulae for those umbers, some special cases ad ew combiatorial idetities are derived Fially, i Sectio 4, a computer program is writte usig Maple ad executed for calculatig the multiparameter o-cetral Stirlig matrix ad some special cases ad matrix represetatio of these umbers is give 2 A modified approach to multiparameter o-cetral Stirlig umbers Let the differetial operator D be defied by D = x α δx α, where δ = xd Usig iii we get D f x = x α δx α f x = δ α f x, hece D = δ α ad by iductio we get D = δ α = x α δ x α Geerally, we ca tae the followig Defiitio 21 Let the differetial operator D be defied by D = x α 1 δx α 1 x α 0 δx α 1 0 = ad D 0 = I, the idetity operator i=0 x α i δx α i, 1 The we have D f x = δ α 1 δ α 0 f x, ie, 1 D = δ α 1 δ α 0 = i=0 δ α i = δ;ᾱ,

4 4 N P Caić, B S El-Desouy ad G V Milovaović 1 ad so we have the operatioal formula x α i δx α 1 i = δ α i i=0 This leads us to defie the multiparameter o-cetral Stirlig umbers via the differetial operator D Therefore, equatios 15 ad 16 ca be represeted, i terms of operatioal formulae, by ad δ = x D = D = δ;ᾱ = s,;ᾱδ;ᾱ = S,;ᾱδ = i=0 s,;ᾱd 21 S,;ᾱx D, 22 respectively Similarly, Comtet umbers see [4] ad [20] ca be defied by ad ad δ;ᾱ = δ = sᾱ,δ Sᾱ,δ;ᾱ Settig α i = α, the 21 ad 22, respectively, yield δ = x D = x α δ x α = δ α = s,;αδ α = S,;αδ = s,;αx α δ x α 23 S,;αx D, 24 where s,;α ad S,;α are the o-cetral Stirlig umbers of the first ad secod id [11], respectively Actig with Eq 23 o e pxr, the multiplyig by e pxr we obtai hece we have, e pxr x D e pxr = x H r x,0, p = 1 s,;αe pxr x α δ x α e pxr, 25 s,;αt α x,r, p, 26 where Hx,α, r p ad T α x, r, p are the geeralized Hermite ad Truesdel polyomials, respectively see [12] ad [16]-[18] Settig r = p = 1 i 25, or 26, we get 1 x = s,;αt α x where T α x are Truesdel polyomials, see [17]

5 Explicit Formulas ad Idetities for Geeralized Stirlig Numbers 5 Similarly, actig with Eq 24 o e pxr, we obtai the iverse relatios T α x,r, p = 1 S,;αx 1 H r x,0, p ad T α x = s,;α 1 x Next, we derive some ew results for Hsu ad Shiue umbers [14] Theorem 22 For the special case α i = r iα/β, i = 0,1,, 1, we have s,;ᾱ = β S 2, 27 Proof From equatio 12 we get β t/βt/β 1t/β 1 = Settig t/β = x, we have β xx 1x 1 = hece x = S 2,t + r α S 2,βx+rβx+r αβx+r 1α, β S 2,x + r/βx + r α/βx + r 1α/β Comparig this equatio with 15 yields 27 Also, it ca be show that S,;ᾱ = β S 1, if α i = iα r/β Furthermore, we show that the geeratig fuctio 14, see [14], of Hsu- Shiue umbers ca be ivestigated from the geeratig fuctio 19, see [11], where α i = r + iβ/α, i = 0,1,, 1 Theorem 23 For the special case α i = r +iβ/α, i = 0,1,, 1, the geeratig fuctio 19 is reduced to the geeratig fuctio 14 ad Proof If we start from 19 S 1, = α s,;ᾱ s,;ᾱ z 0! = 1 + z α j α j

6 6 N P Caić, B S El-Desouy ad G V Milovaović ad settig α j = r/α + jβ/α, j = 0,1,,, we get RHS =,i j = 1 + z r α β α = 1! 1 + z r α β α 1 + z α r + j β α rα + j β α rα + i β α 1 + z j β α i=0,i j where we used the idetity, j j i = 1 + z r α β α = 1 j 1 + z j β α = 1! j i = 1 j j! j! i=0,i j 1 + z α r + j β α β α j i i=0,i j 1 + z j β α 1 j j! j! 1 + z α r, 1 + z β α 1 β α Thus, puttig z = αt i 19 we get s,;ᾱ α t = 1 0!! α 1 + αt r/α 1 + αt β/α 1, β hece by virtue of 14 we have S 1, t 0! = α s,;ᾱ t 0! = αtr/α! where α i = iβ + r/α I fact, this equatio shows that S 1, = α s,;ᾱ, 1 + αt β/α 1, β where α i = iβ + r/α, i = 0,1,, 1 This completes the proof Now, settig α i = iα r/β, i = 0,1,, 1 i 22 we get the followig ew operatioal formula for Hsu-Shiue umbers: 1 i=0 x iα r/β δx iα r/β = β S 1,δ 28 Remar 24 Note that equatios 22 ad 28 for α i = 0, i = 0,1,, 1 ad α = 0, β = 1 ad r = 0, respectively, are reduced to the well ow operatioal represetatio of the ordiary Stirlig umbers of the secod id δ = S,x D

7 Explicit Formulas ad Idetities for Geeralized Stirlig Numbers 7 3 New explicit formulae ad combiatorial idetities We derive ew explicit formulae for both ids of the multiparametar o-cetral Stirlig umbers Theorem 31 The umbers S,;ᾱ, have the followig ew explicit formula α0 1 α1 1 α 1 i 2 S,;ᾱ =, I 1 = i 1 where i j {0,1}, j {0,1,, 1}, ad I 1 := i 1 Proof The statemet for = 0 gives that S,0;ᾱ = 1 α 0 α 1 α 1, which agrees with the defiitio of S,;ᾱ see [11] Also, if i 1 {0,1}, we have that S,;ᾱ = I 2 = 1 1 α0 1 α1 2 α 2 i 3 i 2 α0 + [ 1 α 1 1] i I 2 = α1 2 α 2 i 3 i 2, ie, S,;ᾱ = S 1, 1;ᾱ + α 1 S 1,;ᾱ Therefore, by 18 ad iductio we get the desired result Remar 32 We used α 0 = 0 = 0 Theorem 33 The multiparameter o-cetral Stirlig umbers of the first id have the followig ew explicit expressio i1 + α i1 i2 + α i1 +i s,;ᾱ = 2 1 i + α i1 ++i i ++i = Proof For = 0 we have s,0;ᾱ = α 0 α 0 1α = α 0 31

8 8 N P Caić, B S El-Desouy ad G V Milovaović ad if i {0,1}, we have i1 + α i1 i 1 + α i1 ++i s,;ᾱ = i 1 = i 1 + αi1 ++i α i1 i 1 + α i1 ++i i 1 = 1 1 i 1 = s 1, 1;ᾱ + α + 1s 1,;ᾱ, where + + i 1 = By virtue of 17, this completes the proof It is worth otig that settig α i = r + iβ/α, i {0,1,, 1} i the recurrece relatio 17 for multiparameter o-cetral Stirlig umbers of the first id, we ca get the recurrece relatio 13, ad hece s,;ᾱ = α S 1, Furthermore, Corcio, Hsu ad Ta [9] metioed that the multiparameter o-cetral Stirlig umbers are related with the geeralized Stirlig-type pair of Hsu-Shiue see also [23] by S,;1,α,0 = s,;ᾱ ad S,;α,1,0 = S,;ᾱ for special case α i = iα, i = 0,1,, 1 Corollary 34 A ew explicit expressio for geeralized Stirlig umbers Hsu- Shiue umbers type 13 is give by r/β 1 + r α/β S,;α,β,r = β i0 or i the modified form S,;α,β,r = I 1 = I 1 = 1 + r 1α/β i0 i 2 i 1 r r + β α βi0 r + 2β α βi0 + where we use = 0 r = 0 r + 1β α βi i 2, Proof The proof of this result follows directly from Theorem 31 Namely, by settig α i = iα r/β, i = 0,1,, 1, we get the statemet The previous corollary implies ew explicit expressios for the usual Stirlig umbers of the secod ad first id 1 2 i1 1 i1 i 2 S,;0,1,0 = S, = ++i 1 = i 1 i 1

9 Explicit Formulas ad Idetities for Geeralized Stirlig Numbers 9 ad S,;1,0,0 = s, = respectively ++i 1 = i1 = 1 ++i 1 = i2 1 i i 1 Remar 35 From the above corollary we obtai the explicit formula for the umbers F α,γ, see [7], defied by the triagular recurrece relatio F α,γ + 1, = F α,γ, 1 + γ αf α,γ,, where F α,γ 0,0 = 1 ad F α,γ, = 0 whe < 0, < 0 ad < Namely, for β = 0 ad r = γ, the modified form of S,,α,β,r i Corollary 34 reduces to γ γ α γ 1α F α,γ, = S,;α,0,γ = I 1 = Note that some properties ad idetities o S,;α,β,r caot be obtaied for β = 0, which is a reaso why the explicit formula, ie, the modified form from Corollary 34, is very coveiet to use Corcio et al [8] defied the r,β-stirlig umbers, deoted by β,r usig the followig recurrece relatio + 1 = 1 β,r It is easy to coclude that β,r + β + r β,r β,r i 1 i 1, = S,,0,β,r Thus, we ca obtai a explicit formula for r,β-stirlig umbers from Corollary 34 = S,;0,β,r β,r = I 1 = r/β 1 + r/β i0 1 + r/β i0 i 2 Moreover, El-Desouy [11, Theorem 21] derived the followig explicit formula for s,;ᾱ s,;ᾱ = r= 1 r! r! 1 i r α j r i 1 α j, 32 where, i the secod sum, the summatio exteds over all ordered -tuples of itegers,,i r satisfyig the coditio i r = ad i j 0, j = 1,2,,r

10 10 N P Caić, B S El-Desouy ad G V Milovaović From 31 ad 32, we obtai the followig ew combiatorial idetity i1 + α i1 i2 + α i1 + 1 i + α i1 ++i i = 1 = r= 1 1 r! r! 1 i r α j r α j 1 i Remar 36 Now, it is worth otig that all special cases, i xi, derived i [14], are special cases of the multiparameter o-cetral Stirlig umbers Corcio, Hsu ad Ta [9] proved that S 1, =!α!β 1 j j β/α j + r/a From 33 ad Corollary 34, we have the combiatorial idetity!α! 1 j β/α j + r/a r/β = β j I 1 = 1 + r α/β i r 1α/β i0 i 2 i 1 Hogqua Yu [23, Theorem 4 ad Corollary 5] proved that Sp + l,;β,α,0 0 mod p, 34 where p is a prime umber, ad l are itegers such that l + 1 < < p ad Sp,;α,β,r 0 mod p, 35 where α, β ad r are itegers ad p is a prime, 1 < < p By virtue of 34, 35 ad Corollary 34, we obtai the combiatorial idetities 1 β/α 21 β/α i1 ad α p+l ++i p+l 1 =p+l p + l 11 β/α i1 i p+l 2 β p I p 1 =p i p+l 1 r/β 1 + r α/β i0 p 1 + r p 1α/β i0 i p 2 i p 1 0 mod p, l + 1 < < p, 0 mod p, 1 < < p The degeerate weighted Stirlig umbers, deoted by S,,λ θ, are the pair θ,1,λ see [13]

11 Explicit Formulas ad Idetities for Geeralized Stirlig Numbers 11 Howard [13] derived the followig explicit formula λ + l θ 36 S,,λ θ = 1! l=0 1 +l l Next, we fid ew explicit formulae for the followig special cases of Stirlig umbers Corollary 37 The degeerate weighted Stirlig umbers S,, λ θ have the followig explicit formula λ λ θ + 1 i0 λ + 21 θ i0 S,,λ θ = I 1 =, λ + 11 θ i i 2 i 1 37 Proof The proof follows by settig α = θ, β = 1 ad r = λ i the modified form of Corollary 34 From 36 ad 37 we have the ew combiatorial idetity λ λ θ + 1 i0 λ + 21 θ i0 I 1 = λ + 11 θ i i 2 = 1! l=0 1 +l λ + l θ l Also, settig λ = 0 i 37 see [13, Lemma 21], we have a ew explicit formula for degeerate Stirlig umbers 1 θ 21 θ i1 S, θ = ++i 1 = ++i 1 = i 1 11 θ i i 2 i 1 I the special case θ = λ = 0 we obtai the idetity 1 2 i1 1 i i 2 From 11 we get α t/αt/α 1t/α 1 = i 1 = 1! l=0 1 +l l l S 1,t r β

12 12 N P Caić, B S El-Desouy ad G V Milovaović Settig t/α = x, α xx 1x 1= hece x = S 1,αx rαx r βαx r 1β, α S 1,x r/α β/α 38 From 11 ad 38, we obtai α S 1, = S,;1,r/α,β/α, whece Similarly, we ca prove that S,;α,β,r = α S,;1,β/α,r/α 39 S,;α,β,r = β S,;α/β,1,r/β 310 Ideed 39 ad 310 see [1] ca be ivestigated from 14 Usig 310, as well as the fact that the degeerate weighted Stirlig umbers S,,λ θ are the pair θ,1,λ, the the umbers S,;α,β,r ca be represeted i terms of S,,λ θ where S,;α,β,r = β S,,r/β α/β 311 Remar 38 I fact, 311 agrees with Corollary 34 Furthermore, Muagi [15] defied S b,, the B-Stirlig umbers of the secod id, by S b, = S b 1, b 1S b 1,,, S b,0 = δ,0, S b,1 = b 1, where δ i, j is the Kroecer delta Corollary 39 The B-Stirlig umbers of the secod id, S b,, have the followig ew explicit formula S b + 1, + 1 = S,;0,1,b b b + 1 i0 b + 2 i0 + = I 1 = b + 1 i i 2, 312 where b 0 ad we use = 0 b = 0 Proof The proof follows by settig α i = b, i = 0,1,, 1, i Theorem 31 or settig α = 0, β = 1 ad r = b i Corollary 34 This show that the B-Stirlig umbers of the secod id is a special case of the multiparameter o-cetral Stirlig umbers ad cosequetly of Hsu-Shiue umbers i 1

13 Explicit Formulas ad Idetities for Geeralized Stirlig Numbers 13 From 312 ad [15, Corollary 62] we obtai the ew combiatorial idetity b b + 1 i0 b + 1 i i 2 I 1 = i 1 = 1 j b + j j! j! 4 The matrix represetatio Let s,s; sα,sα ad S 1,S 2 be lower triagular matrices, where s ad S are the matrices whose etries are the Stirlig umbers of the first ad secod ids ie, s = [s i j ] i, j 0 ad S = [S i j ] i, j 0 ; sα ad Sα are the matrices whose etries are the multiparameter o-cetral Stirlig umbers of the first ad secod ids ie, sα = [sα i j ] i, j 0 ad Sα = [Sα i j ] i, j 0 ad S 1,S 2 are the matrices whose etries are the geeralized Stirlig-type pair of Hsu ad Shiue ie, S 1 = [S 1 i j ] i, j 0 ad S 2 = [S 2 i j ] i, j 0, respectively The multiparameter o-cetral Stirlig umbers of the first ad secod ids, Eqs 21 ad 22, ca be represeted i a matrix form as δ = sα D ad D = Sα δ, respectively, where δ = δ 0,δ 1,,δ T ad D = D 0, D 1,,D T A computer program is writte usig Maple ad executed for calculatig the multiparameter o-cetral Stirlig matrix of the first id sα ad Sα = s 1 α of the secod id ad the geeralized Stirlig-type pair of Hsu ad Shiue as a special case whe α i = r + iβ/α For example if 0 3, the sα = α α 0 α 0 1 α 0 + α α 0 α 0 1α 0 2 α 0 α 1 3α 0 + α0 2 3α 1 + α α 0 + α 1 + α ad S 1 = r rr α 2r + β α 1 0 rr αr 2α 3r 2 6rα + 3βr + β 2 3βα + 2α 2 3r + 3β 3α 1 Remar 41 Note that there are some missig terms i [23, Table 1] Similarly, the degeerate weighted Stirlig matrix Sλ θ, the degeerate Stirlig matrix Sθ ad the Stirlig matrix of the secod id S are give by Sλ θ = λ λλ θ 2λ + 1 θ 1 0, λλ θr 2θ 3λ 2 6λθ + 3λ + 1 3θ + 2θ 2 3λ + 3 3θ 1

14 14 N P Caić, B S El-Desouy ad G V Milovaović ad Sθ = θ θ1 2θ 31 θ 1 S = respectively, as special cases Acowledgemets The authors would lie to tha the aoymous referee for helpful commets ad suggestios,, Refereces [1] D Braso, Stirlig umbers ad Bell umbers: their role i combiatorics ad probability Math Sci , 1 31 [2] Whei-Chig C Cha, Kug-Yu Che ad H M Srivastava, Certai classes of geeratig fuctios for the Jacobi ad related hypergeometric polyomials Comput Math Appl , [3] Wechag Chu ad Chuaam Wei, Set partitios with restrictios Discrete Math , [4] L Comtet, Nombres de Stirlig geeraux et foctios symetriques C R Acad Sci Paris Ser A , [5] L Comtet, Advaced Combiatorics: The Art of Fiite ad Ifiite Expasios D Reidel Publishig Compay, Dordrecht, Holad, 1974 [6] R B Corcio, Some theorems o geeralized Stirlig umbers Ars Combi [7] R B Corcio ad M Herrera, Limit of the differeces of the geeralized factorial, its q- ad p,q-aalogue Util Math , [8] R B Corcio, C B Corcio ad R Aldema, Asymptotic ormality of the r,β-stirlig umbers Ars Combi , [9] R B Corcio, L C Hsu ad E L Ta, A q-aalogue of geeralized Stirlig umbers Fiboacci Quart , [10] G Dattoli, M X He, ad P E Ricci, Eigefuctios of Laguerre-type operators ad geeralized ievolutio problems Math Comput Modellig , [11] B S El-Desouy, The multiparameter o-cetral Stirlig umbers Fiboacci Quart , [12] H W Gould ad A T Hopper, Operatioal formulas coected with two geeralizatios of Hermite polyomials Due Math J , [13] F T Howard, Degeerate weighted Stirlig umbers Discrete Math , [14] L C Hsu ad P J-Sh Shiue, A uified approach to geeralized Stirlig umbers Adv i Appl Math , [15] A O Muagi, Labeled factorizatio of itegers Electro J Combi , R50

15 Explicit Formulas ad Idetities for Geeralized Stirlig Numbers 15 [16] P N Shrivastava, O the polyomials of Truesdel type Publ Ist Math Beograd NS , [17] R P Sigh, O geeralized Truesdel polyomials Riv Mat Uiv Parma, , [18] H M Srivastava ad H L Maocha, A Treatise o Geeratig Fuctios Ellis Horwood Limited, Chichester, 1984 [19] O V Visov ad H M Srivastava, New approaches to certai idetities ivolvig differetial operators J Math Aal Appl , 1 10 [20] C G Wager, Geeralized Stirlig ad Lah umbers Discrete Math , [21] C G Wager, Partitio statistics ad q-bell umbers q = 1 J Iteger Seq , Article 0411 [22] H S Wilf, Geeratigfuctioology Academic Press Ic, New Yor, 1994 [23] H Yu, A geeralizatio of Stirlig umbers Fiboacci Quart , Nead P Caić Departmet of Mathematics Faculty of Electrical Egieerig Uiversity of Belgrade PO Box Belgrade Serbia caic@etfrs Beih S El-Desouy Departmet of Mathematics Faculty of Sciece Masoura Uiversity Masoura Egypt b desouy@yahoocom Gradimir V Milovaović 1 Mathematical Istitute Serbia Academy of Sciece ad Arts Keza Mihaila 36 PO Belgrade Serbia gvm@misauacrs 1 Correspodig author