A solid Foundation for q-appell Polynomials

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1 Advaces i Dyamical Systems ad Applicatios ISSN , Volume 10, Number 1, pp ) A solid Foudatio for -Appell Polyomials Thomas Erst Uppsala Uiversity Departmet of Mathematics P.O. Box 480, SE Uppsala, Swede thomas@math.uu.se Abstract The term multiplicative -Appell polyomial is itroduced, with immediate iterestig geeralizatios for -aalogues of Nørluds formulas. We show that the set of -Appell polyomials is a commutative rig. Fially, we show some formulas for -Geroimus Appell polyomials, geeralizig a paper of Geroimus from AMS Subject Classificatios: 05A40, 11B68, 05A10. Keywords: Multiplicative -Appell polyomial, umbral calculus, commutative rig. 1 Itroductio The purpose of this article is to geeralize the previous expasio formulas from [5] to multiplicative -Appell polyomials. The multiplicative -Appell polyomials possess a certai multiplicative coditio o the geeratig fuctio. We will also put the calculatios for -Appell polyomials o a solid algebraic basis, establishig a li betwee algebra ad special fuctios. May calculatios o Appell polyomials today are made without owig that they are, i fact, Appell polyomials. This meas that our article will also be of great use for scietists worig with the udeformed case = 1. This paper is orgaized as follows: I this sectio we give a short itroductio ad mae some defiitios. I sectio two we show that the -Appell polyomials ad the -Appell umbers as well) form a commutative rig with two operatios. ad, iduced from + ad. I sectio three we discuss multiplicative -Appell polyomials. We will defie three ids of multiplicative -Appell polyomials, for the purpose of defiig NWA ad JHC triagle operators, ad derivig complemetary argumet theorems ad Nørlud [7] type expasio formulas. Some of the proofs are the same Received February 19, 2014; Accepted December 17, 2014 Commuicated by Lace Littlejoh

2 28 Thomas Erst as i [1]. I a appedix we itroduce the very geeral -Geroimus Appell polyomials, which cotai a extra variable y, ad show some remarable symmetry properties. We ow start with the defiitios, compare with the boo [2]. Some of the otatio is well-ow ad will be sipped. Defiitio 1.1. Let the Gauss -biomial coefficiet be defied by {}!, = 0, 1,...,. 1.1) {}!{ }! Let a ad b be ay elemets with commutative multiplicatio. The the NWA -additio is give by a b) a b, = 0, 1, 2, ) =0 If 0 < < 1 ad z < 1 1, the -expoetial fuctio is defied by 1 E z) {}! z. 1.3) The -derivative is defied by fx) fx), if C\{1}, x 0; 1 )x D f) x) df x) if = 1; dx df 0) if x = 0. dx =0 Theorem 1.2. We have the -Taylor formula ν ν ν, x y) = =0 ) 1.4) ν, x)y. 1.5) Defiitio 1.3. For every power series f t), the -Appell polyomials or Φ polyomials of degree ν ad order have the followig geeratig fuctio: t ν f t)e xt) = {ν}! Φ) ν, x). 1.6) For x = 0 we get the ν, umber of degree ν ad order. Defiitio 1.4. Uder the assumptio that the fuctio ht) ca be expressed aalytically i R[[t]], ad for f t) of the form ht), we call the -Appell polyomial Φ i 1.6) multiplicative. E xt)ht) = t ν {ν}! Φ) M,ν, x), N. 1.7)

3 -Appell Polyomials 29 The we have Theorem 1.5. If s l =, N, l=1 M,, x 1... x s ) = m m s= where we assume that j operates o x j. m 1,..., m s ) s j=1 Φ j) M,m j, x j), 1.8) Two special cases of multiplicative -Appell polyomials are NWA multiplicative -Appell polyomials NWA,M,ν, x) ad JHC multiplicative -Appell polyomials JHC,M,ν, Defiitio 1.6. x), which are defied as follows: E xt)h 1 t) = t ν {ν}! Φ) NWA,M,ν, x), N, 1.9) where the oly -expoetial fuctio i h 1 t) is E t). Defiitio 1.7. E xt)h 2 t) = t ν {ν}! Φ) JHC,M,ν, x), N, 1.10) where the oly -expoetial fuctio i h 2 t) is E 1 t). We assume that NWA,M,ν, x) ad Φ) JHC,M,ν, x) always occur together. The fuctio h 2 t) is eual to a iversio of the basis i h 1 t). Defiitio 1.8. Assume that Z. We defie the operator M, by M, M,ν, x) = D Φ 1) M,ν, x). 1.11) The operators M, ad D commute sice D maps a -Appell polyomial to aother -Appell polyomial.

4 30 Thomas Erst 2 The Rig of -Appell Polyomials Defiitio 2.1. We deote the set of all -Appell polyomials i the variable x) by A x;. Let Φ, x) ad Ψ, x) be two elemets i A x;. The the operatios ad are defied as follows: Φ x) Ψ x)) Φ x) + Ψ x)), 2.1) Φ x) Ψ x)) Φ x) Ψ x)) = Φ, x)ψ, x). 2.2) We eep the usual priority betwee ad. =0 Theorem 2.2. A x;, ) is a commutative rig. Proof. We presume that Φ, x), Ξ, x) ad Ψ, x) are three elemets i A correspodig to the geeratig fuctios ft), gt) ad ht) respectively. We first show that is well-defied. Assume that f0) + h0) 0. The Φ x) Ψ x) A x;, 2.3) Φ, x) Ψ, x) has geeratig fuctio ft) + ht). 2.4) The associative law for reads: Φ x) Ξ x) ) Ψ x)) = Φ x) Ξ x)) ) Ψ x). 2.5) This follows from the associativity of +. The commutative law for reads: Φ x) ) Ψ x) = Ψ x) ) Φ x). 2.6) This follows from the commutativity of +. The idetity elemet with respect to is the seuece O. We have Φ x) ) O = Φ x)). 2.7) There exists Φ x) such that Φ x) ) Φ x)) = O). 2.8)

5 -Appell Polyomials 31 This follows from the correspodig property of real umbers. The we show that is well-defied. We have Φ x) Ψ x) A x;, 2.9) Φ, x) Ψ, x) has geeratig fuctio ft)ht). 2.10) This follows sice i the umbral sese ft)ht) = E tφ Ψ )). 2.11) The idetity elemet with respect to is I δ,0. This correspods to ft) = 1. To show this, we put = ad Φ = 1 i 2.2). We have The associative law for reads: Φ x) ) I = I ) Φ x) = Φ, x). 2.12) Φ x) Ξ x) ) Ψ x)) = Φ x) Ξ x)) ) Ψ x). 2.13) This follows from the associativity of. The commutative law for reads: Φ x) ) Ψ x) = Ψ x) ) Φ x). 2.14) This follows from the commutativity of. The distributive law reads: Φ x) Ξ x) ) Ψ x)) = Φ x) Ξ x) Φ, x) ) Ψ x). 2.15) This follows from the followig computatio: Φ x) Ξ x) ) Ψ x)) by2.2)) = by2.1)) = =0 =0 Φ, x) Ξ x) Ψ x) Φ, x) Ξ x) + Ψ x)). ) 2.16)

6 32 Thomas Erst O the other had, Φ x) Ξ x) Φ x) ) Ψ x) by2.1)) = Φ x) Ξ x) + Φ x) ) Ψ x) by2.2)) = Φ, x) Ξ x) + Ψ x)). =0 2.17) Both expressios are eual. Corollary 2.3. The set of all -Appell umbers A is a commutative rig. It is also a subrig of A x;. Proof. Put x = 0 i the previous theorem. 3 Multiplicative -Appell Polyomials Assume that, p Z. The all the followig formulas from [2] also hold for geeral multiplicative -Appell polyomials. Theorem 3.1. Φ p) M,ν, x y) =Φ ) M, x) Φ p) M, y))ν. 3.1) A special case is the followig formula: Φ ) M,ν, x y) =Φ ) M, x) y) ν. 3.2) Theorem 3.2. A recurrece formula for the NWA multiplicative -Appell umbers: If, p Z the Φ +p) M,ν, =Φ) M, Φ p) M, )ν, 3.3) Theorem 3.3. x y) ν =Φ ) M, x) M, y))ν, 3.4) Proof. Put p = i 3.1). I particular for y = 0, we obtai x ν =Φ ) M, M, x))ν. 3.5) These recurrece formulas express NWA multiplicative -Appell polyomials of order without metioig polyomials of egative order.

7 -Appell Polyomials 33 This ca be expressed i aother form: x ν = ν s=0 Φ ) M,s, {s}! Ds M,ν, x), 3.6) We coclude that NWA multiplicative -Appell polyomials satisfy liear -differece euatios with costat coefficiets. The followig formula is useful for the computatio of multiplicative -Appell polyomials of positive order. This is because the polyomials of egative order are of simpler ature ad ca easily be computed. Whe the Φ ) M,s, used to compute the M,s,. Theorem 3.4. ν s=0 Proof. Put x = y = 0 i 3.4). etc. are ow, 3.7) ca be ) ν M,s, s Φ ) M,ν s, = δ ν,0. 3.7) Theorem 3.5. A geeralizatio of [2, 4.261]: Uder the assumptio that fx) is aalytic with -Taylor expasio fx) = D ν f0) xν {ν}!, 3.8) we ca express powers of M, operatig o fx) as powers of D as follows. These series coverge whe the absolute value of x is small eough: M,fx) = Proof. Simply use formula 1.11). D ν+ f0) Φ ) M,ν, x) {ν}!, > ) 4 Appedix: The Geroimus Appell Polyomials Yaov Lazarevich Geroimus ) obtaied his thesis from Charov State Uiversity i 1939 o the subject approximatios ad expasios. His tutor was the famous Sergei Nataovich Berstei ), well ow for the Berstei polyomials. To explai the coectio with Appell polyomials we have to go bac to Berstei s studet years i Paris. Berstei submitted his thesis at Sorboe i the sprig of 1904, supervised by Hilbert ad Picard. Probably Berstei leared about Appell polyomials durig this time, ad he passed o this owledge to Geroimus. He could also have bee iflueced by his coutryma Vashcheo-Zaharcheo ), who had

8 34 Thomas Erst studied umbral calculus i Paris [2, p. 69]. We are ow goig to fid the - aalogues of the first formulas i Geroimus article [6]. We first defie a very geeral -differece operator. Defiitio 4.1. Let hx.) be a fuctio of two variables. The operator x,; is defied by ) ) ) x x x,h; hx.) = hx. 1). 4.1) 1 Defiitio 4.2. A -aalogue of [6, p. 13, 1)]. The geeratig fuctio for the - Geroimus Appell polyomials Ω ν, x, y, f) is give by ft) a give fuctio ft)e xt) ) y t l hy.l) = l l=0 ft) = =0 We obtai the fudametal property Theorem 4.3. A -aalogue of [6, p. 13, 4)] Proof. Operate with y,h; o 4.2). t ν {ν}! Ω ν,x, y, f), 4.2) f t {}!, f costats. 4.3) y,h; Ω ν, x, y, f) = {ν} Ω ν 1, x, y, f). 4.4) We observe that Ω ν, x, y, f) cotais a extra variable y. The relatio 4.4) is very similar to the defiig relatio for -Appell polyomials. Corollary 4.4. A -aalogue of [6, p. 13, 6)] 5 Discussio Ω, x, y, f) {}! = =0 ) y hy,) x f). 4.5) { }! We have geeralized some of the previous formulas from [1] ad [2] to multiplicative -Appell polyomials. This calculus should be adapted to special cases of Appell polyomials, which will hopefully appear i the future. The fact that -Appell polyomials is a rig puts the subject o a solid mathematical basis ad improves the characterizatio i [3]. I fact this is a special case of the matrix -expoetial fuctio for the -Polya-Vei matrix [4, p. 1178]).

9 -Appell Polyomials 35 Refereces [1] Erst T., -Beroulli ad -Euler Polyomials, A Umbral Approach. Iteratioal Joural of Differece Euatios 1, o ), [2] Erst T., A comprehesive treatmet of -calculus, Birhäuser [3] Erst T., -Pascal ad -Wrosia matrices with implicatios to -Appell polyomials. J. Discrete Math [4] Erst T., A umbral approach to fid -aalogues of matrix formulas. Liear Algebra Appl. 439, o ), [5] Erst T., O certai geeralized -Appell polyomial expasios. Submitted. [6] Geroimus Y., O a class of appell polyomials. Commu. Soc. Math. Kharoff et Ist. Sci. Math. et Meca., Uiv. Kharoff, IV. Ser ) [7] Nørlud N.E., Differezerechug. Berli 1924.