SOME IDENTITIES FOR A SEQUENCE OF UNNAMED POLYNOMIALS CONNECTED WITH THE BELL POLYNOMIALS FENG QI

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1 SOME IDENTITIES FOR A SEQUENCE OF UNNAMED POLYNOMIALS CONNECTED WITH THE BELL POLYNOMIALS FENG QI Istitute of Mathematics, Hea Polytechic Uiversity, Jiaozuo City, Hea Provice, 45400, Chia; College of Mathematics, Ier Mogolia Uiversity for Natioalities, Togliao City, Ier Mogolia Autoomous Regio, 08043, Chia; Departmet of Mathematics, College of Sciece, Tiaji Polytechic Uiversity, Tiaji City, , Chia DA-WEI NIU Departmet of Mathematics, East Chia Normal Uiversity, Shaghai City, 004, Chia BAI-NI GUO School of Mathematics ad Iformatics, Hea Polytechic Uiversity, Jiaozuo City, Hea Provice, 45400, Chia Abstract I the paper, usig two iversio theorems for the Stirlig umbers ad biomial coefficiets, employig properties of the Bell polyomials of the secod id, ad utilizig a higher order derivative formula for the ratio of two differetiable fuctios, the authors preset two explicit formulas, a determiatal expressio, ad a recursive relatio for a seuece of uamed polyomials, derive two idetities coectig the seuece of uamed polyomials with the Bell polyomials, ad recover a ow idetity coectig the seuece of uamed polyomials with the Bell polyomials addresses: ifeg68@gmailcom, ifeg68@hotmailcom, ifeg68@com, ddww@gmailcom, baiiguo@gmailcom, baiiguo@hotmailcom 00 Mathematics Subject Classificatio Primary B83; Secodary 05A5, A5, B65, B73, C08, 33B0 Key words ad phrases idetity; Bell polyomial; uamed polyomial; explicit formula; iversio theorem; Stirlig umber; biomial coefficiet This paper was typeset usig AMS-LATEX 07 by the authors Distributed uder a Creative Commos CC BY licese

2 F QI, D-W NIU, AND B-N GUO Motivatios I [4, pp 57 58] ad [5], the fuctios i [ ] F t, x = exp x = h x t t t! =0 were cosidered I [4, pp 57 58], it was poited out that the uamed polyomials h x satisfy h x = [ ] x d x! e x dx e x I [5], it was poited out that the uamed polyomials h x ad the Bell polyomials B x are coected by the idetity B x = h xs,, 0, 3 =0 =0 where the Bell polyomials B x ca be geerated by e xet = B x t! ad the Stirlig umbers of the secod id S, ca be geerated by e x = S, x, {0} N!! = =0 It was poited out i [4, pp 57 58] that the expressio had bee applied i 937 to the theory of hyperbolic differetial euatios It was also poited out i [0] that there have bee some studies o iterestig applicatios of Bell polyomials B x i solito theory, icludig lis with biliear ad triliear forms of oliear differetial euatios which possess solito solutios See, for example, [6, 7, 8] Therefore, applicatios of the Bell polyomials B x to itegrable oliear euatios are greatly expected ad ay amedmet o multi-liear forms of solito euatios, eve o exact solutios, would be beeficial to iterested audieces i the research commuity To simplify mai results i [5], amog other thigs, the followig coclusios were established i the ewly published paper [0] Theorem [0, Theorem ] For 0, the th derivative of the geeratig fuctio F t, x of the polyomials h x i ca be computed by [ d F t, x F t, x ] l l x d t = l! l!! l! t, +/ =0 l= where the double factorial of egative odd itegers + is defied by!! =!! =!!, 0 ad the covetios 0 0 = ad p = 0 for > p 0 are adopted Coseuetly, the polyomials h x for 0 ca be expressed as h x = [ l l! l!! l =0 l= l ] x!

3 IDENTITIES FOR A SEQUENCE OF UNNAMED POLYNOMIALS 3 Theorem [0, Theorem 3] For 0, the Bell polyomials B x ad the polyomials h x ca be expressed each other by [ ] B x = l S, l h l x 4 ad h x = =l =l [ ]!! l s, l B l x, 5 where s, for 0, which ca be geerated by [l + x] = s, x, x <,!! = stad for the Stirlig umbers of the first id Theorem 3 [0, Theorem 4] For 0, the Bell polyomials B x ad the polyomials h x satisfy the idetities 3!!h x = s, B x 6 ad =0 =0 =0 [ ] s, l!! l B l x h x = 7 I this paper, usig two iversio theorems for the Stirlig umbers s, ad S, ad biomial coefficiets, employig properties of the Bell polyomials of the secod id B,, ad utilizig a higher order derivative formula for the ratio of two differetiable fuctios, the authors preset two explicit formulas, a determiatal expressio, ad a recursive relatio for h x, derive two idetities coectig h x with B x, ad recover the idetity 3 Our mai results ca be stated as the followig four theorems Theorem 4 For 0, the polyomials h x satisfy h x =! l l [ l ]!! x l, l where =0 is the fallig factorial x = x = =0 =0 { xx x +,, = 0 Theorem 5 For 0, the Bell polyomials B x ad h x satisfy B x = S, l l 3!!h l x 8 l =0 Theorem 6 For 0, the idetity 3 is valid ad s, h x = B l x 9 l

4 4 F QI, D-W NIU, AND B-N GUO Theorem 7 For 0, the polyomials h x ca be computed by the determiatal expressio λ 0 x 0 0 0!! λ x λ 0 x 0 0 3!! λ x λ x 0 0 h x = 5!! λ x, λ 3 x λ 0 x 0 3!! λ x λ x λ x λ 0 x!! λ x λ x λ x λ x 0 by the recursive relatio!! λ r xh r x = r, r=0 ad by the explicit formula for 0, where λ x = h x =! =0 [ l l p p p=0 [ ]!! λ x [ ]!!! ] x l p + l!, 0 Lemmas I order to prove our mai results, we recall several lemmas below Lemma [, Theorem 4] For 0, we have d e u d u = e u u!! u / =0 Lemma [4, p 7, Theorem ] If b α ad a are a collectio of costats idepedet of, the a = S, αb α if ad oly if b = s, a α=0 Lemma 3 [4, p 83, E 7] If a ad b for 0 are a collectio of costats idepedet of such that 0, the a = b if ad oly if b = a =0 Lemma 4 [3, pp 34 ad 39] For 0, the Bell polyomials of the secod id, or say, partial Bell polyomials, deoted by B, x, x,, x +, are defied by B, x, x,, x + = i,l i {0} N i= ili= i= li= =0 =0! + i= l i! + i= xi i! li

5 IDENTITIES FOR A SEQUENCE OF UNNAMED POLYNOMIALS 5 The Faà di Bruo formula ca be described i terms of the Bell polyomials of the secod id B, x, x,, x + by d d t f ht = =0 Lemma 5 [3, p 35] For 0, we have f htb, h t, h t,, h + t B, abx, ab x,, ab + x + = a b B, x, x,, x +, where a ad b are ay complex umbers Lemma 6 [9, Remar ] For 0, we have B,, λ, λ λ,, lλ =! l l l λ 3 Lemma 7 [, p 40, Etry 5] Let p = px ad = x 0 be two differetiable fuctios The p p 0 0 [ ] p 0 0 px = x + p p p for 0 I other words, the formula 4 ca be rewritte as d [ ] px d x = W+ + x + x, 5 x where W + + x deotes the determiat of the + + matrix W + + x = U + x V + x, the uatity U + x is a + matrix whose elemets u l, x = p l x for l +, ad V + x is a + matrix whose elemets for i + ad j i i j x, i j 0 v i,j x = j 0, i j < 0

6 6 F QI, D-W NIU, AND B-N GUO Lemma 8 [, p, Theorem] ad [, Remar 3] Let M 0 = ad m, m, m, m, m,3 0 0 m 3, m 3, m 3,3 0 0 M = m, m, m,3 m, 0 m, m, m,3 m, m, m, m, m,3 m, m, for N The the seuece M for 0 satisfies M = m, ad M = m, M + r m,r m j,j+ M r, 6 r= j=r 3 Proofs of mai results Now we are i a positio to prove our mai results Proof of Theorem 4 By virtue of Lemma, the formula implies that h x = x d u /! e x e u, u = x d u = x u / e u! e x =0 = x! e x / u =0 u e u l l [ l ]!! u l/ l =! l l [ l ]!! x l l =0 The proof of Theorem 4 is complete Proof of Theorem 5 The idetity 6 i Theorem 3 ca be rearraged as 3!!h x = s, B x =0 Further utilizig Lemma yields B x = S, =0 The proof of Theorem 5 is complete =0 l l 3!!h l x l Proof of Theorem 6 Iterchagig the order of sums i the right had side of the idetity 4 i Theorem arrives at B x = S, l l h l x =0

7 IDENTITIES FOR A SEQUENCE OF UNNAMED POLYNOMIALS 7 Further applyig Lemma 3 leads to S, l l h l x = B x = =0 which is euivalet to 3 Employig Lemma i 3 produces h x = =0 The proof of Theorem 6 is complete s, B l x l =0 B x Proof of Theorem 7 By straightforward computatio, we obtai d [ t / ] d t = t /!! = =, t 0 ad, whe deotig u = ut = t / ad maig use of the formulas ad, d exp [ x t /] d t = e xu l B,l u t, u t,, u l+ t = x l e xu B,l t, 3/ t,, 5/ l+ l+ t l+3/!! e x x l B,l, 3!! l +!!,, l+, t 0 = e x x l B,l!!, 3!!,, l +!! Taig λ = i 3 gives B,!!, 3!!,, +!! =! for 0 Therefore, we obtai d exp [ x t /] lim t 0 d t = e x The euatio meas that h x = lim t 0 x l l! l l l l p p p=0 d d t t / exp [ x t /] = e x lim t 0 l + p + = e x λ x d d t t / exp [ x t /]

8 8 F QI, D-W NIU, AND B-N GUO Further applyig the formula 5 to results i pt = t / ad t = exp [ x t /] h x = e x lim exp [ + x t /] t 0 p p 0 0 p 0 0 lim t 0 p 3 0 p p 0 x 0 0 0!! x 0 x 0 0 3!! x x 0 0 = e x 5!! x 3 x 0 x 0 3!! x x x 0 x!! x x x λ 0 x 0 0 0!! λ x λ 0 x 0 0 3!! λ x λ x 0 0 = 5!! λ x λ 3 x λ 0 x 0 3!! λ x λ x λ x λ 0 x!! λ x λ x λ x, x λ x where x = e x λ x for 0 The determiatal expressio 0 is proved Applyig 6 to the determiatal expressio 0 ad cosiderig λ 0 x = derive h x = λ x!! h x + + r+ λ r+ x r h r x, r r= This ca be reformulated as From the euatio ad the above argumets, it follows that h x = e x d [ ] x lim t 0 d t exp t t d = e x d x lim t 0 d t t d t exp t =0

9 = e x = IDENTITIES FOR A SEQUENCE OF UNNAMED POLYNOMIALS 9 = e x =0 =0 =0 lim t 0!! e x!! d d t t lim t 0 x l l! l l! x l d x d t exp t l l p p p=0 l l p p p=0 p + p + which ca be rewritte as The proof of Theorem 7 is complete 4 Remars Fially we list several remars about our mai results ad other thigs Remar 4 Recetly a ew iversio theorem was discovered i [3, Theorem 43] which ca be reformulated as that s = S if ad oly if S = s, = where s ad S are two seueces idepedet of such that = Remar 4 The idetity 4 is slightly differet from 8 Remar 43 The idetity 9 is obviously simpler tha 5 i their forms Remar 44 Comparig 7 with reveals that that is, s, l l B l x = λ x, Further cosiderig Lemma deduces s, l B lx l = λ x B x = Refereces S, lλ l x [] N Bourbai, Fuctios of a Real Variable, Elemetary Theory, Traslated from the 976 Frech origial by Philip Spai Elemets of Mathematics Berli Spriger-Verlag, Berli, 004; Available olie at [] N D Cahill, J R D Errico, D A Naraya, ad J Y Naraya, Fiboacci determiats, College Math J 3 00, 5; Available olie at [3] L Comtet, Advaced Combiatorics: The Art of Fiite ad Ifiite Expasios, Revised ad Elarged Editio, D Reidel Publishig Co, 974 [4] A Erdélyi, W Magus, F Oberhettiger, ad F G Tricomi, Higher Trascedetal Fuctios, Vol III Based o otes left by Harry Batema Reprit of the 955 origial Robert E Krieger Publishig Co, Ic, Melboure, Fla, 98 [5] T Kim, D V Dolgy, D S Kim, H I Kwo, ad J J Seo, Differetial euatios arisig from certai Sheffer seuece, J Comput Aal Appl 3 07, o 8, [6] W-X Ma, Biliear euatios, Bell polyomials ad liear superpositio priciple, J Phys Cof Ser 4 03, o, Aricle ID 00, pages; Available olie at org/0088/ /4//00

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