Ruscheweyh and Salinas showed in 2004 the relationship of a celebrated theorem of Vietoris (1958) about the positivity of certain sine and cosine sums with the function theoretic concept of stable holomorphic functions in the unit disc. The present paper reveals that the coefficient sequence in Vietoris' theorem is identical to a number sequence obtained by a new combinatorial identity which involves  generators of quaternions. In this sense Vietoris' sequence of rational numbers combines seemingly disperse subjects in Real, Complex and Hypercomplex Analysis. Thereby we show that a non-standard application of Clifford algebra tools is able to reveal new insights in objects of combinatorial nature.The work of the first and third authors was supported by Portuguese funds
through the CIDMA - Center for Research and Development in Mathematics
and Applications, and the Portuguese Foundation for Science and Technology
(“FCT-Fundação para a Ciência e Tecnologia”), within project PEstOE/MAT/UI4106/2013.
The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio

Cação, I.

Falcão, M. I.

Malonek, H. R.

Universidade do Minho: RepositoriUM

On Vietoris’ number sequence and combinatorialidentities with quaternionsI. Cac¸a˜oa M. I. Falca˜ob H. R. Malonekaisabel.cacao@ua.pt mif@math.uminho.pt hrmalon@ua.pta CIDMA and Department of Mathematics, University of Aveiro, Aveiro, Portugalb CMAT and Department of Mathematics and Applications, University of Minho,Campus de Gualtar, Braga, PortugalAbstractRuscheweyh and Salinas showed in 2004 the relationship of a cele-brated theorem of Vietoris (1958) about the positivity of certain sine andcosine sums with the function theoretic concept of stable holomorphicfunctions in the unit disc. The present paper reveals that the coefficientsequence in Vietoris’ theorem is identical to a number sequence obtainedby a new combinatorial identity which involves generators of quaternions.In this sense Vietoris’ sequence of rational numbers combines seeminglydisperse subjects in Real, Complex and Hypercomplex Analysis. Therebywe show that a non-standard application of Clifford algebra tools is ableto reveal new insights in objects of combinatorial nature.Keywords: Vietoris’ number sequence, quaternions, combinatorial identities1 IntroductionIn the center of our attention lies the sequence of rational numbers1, 12 ,12 ,38 ,38 ,516 ,516 ,35128 ,35128 ,63256 ,63256 ,2311024 ,2311024 , . . . . (1)which by means of the generalized central binomial coefficient( kbk2 c)can bewritten in compact form (cf. [4]) asS = (ck)k≥0 , where ck =12k(kbk2 c), k ≥ 0. (2)Seemingly for the first time this sequence appeared in the context of positivetrigonometric sums in a celebrated paper of L. Vietoris [17]. Askey’s version([2, p. 5]) of Vietoris’ theorem is the following:Theorem 1 (L. Vietoris)n∑k=1ak sin kθ > 0, 0 < θ < pi, andn∑k=0ak cos kθ > 0, 0 ≤ θ < pi,1wherea2k = a2k+1 =( 12 )kk!, k = 0, 1, . . . , (3)with (·)k as the raising factorial in the classical form of the Pochhammer symbol.We call attention to the fact that because of (3), the coefficients in the sine sumused in Askey’s as well as in Vietoris’ original version are exactly the elementsof S in (2) or, explicitly, in (1). Obviously, demanding in (3) that a2k anda2k+1 coincide, the sequence of coefficients in the cosine sum differs from (1)by the inclusion of a0 = 1 and the shift of the indices by one to the left, i.e.a0 = 1 and ak+1 = ck, k ≥ 0. Even though this small difference, we call Sin the sequel simply Vietoris’ number sequence. Compared with the traditionalway of defining the coefficient sequence by (3), the use of the properties of thegeneralized central binomial coefficient allowed at least a unique representation(2) with consecutively running index k.Before continuing with the specific task of the present paper, it seems worth-while to mention the other areas in which Vietoris’ theorem played an importantrole. Using the arsenal of real analysis methods in positivity theory, Askey andSteinig showed in [3] the embedding of Vietoris’ results in general problems forJacobi polynomials, including their relation to other subjects in Harmonic Anal-ysis. Later on, Ruscheweyh and Salinas showed in [15] an interesting relationshipof Vietoris’ theorem with the function theoretic concept of stable holomorphicfunctions in the unit disc. The common origin of the present paper with otherslike, for example, [9, 5] where the sequence S was already mentioned in differ-ent contexts, is the field of Hypercomplex Analysis, particularly the study ofmonogenic (or Clifford-holomorphic) Appell polynomials [1, 8, 10]. Recently in[6], the authors obtained even some number theoretic results for a related to Sinteger number sequence (sequence A283208 in The On-Line Encyclopedia ofInteger Sequences, published electronically at https://oeis.org).The goal of the present paper concerns the surprising appearance of S ina relation between the generators of Hamilton’s well known non-commutativealgebra H of quaternions (see e.g. [10]), relying only on elementary proper-ties of a adequately generalized binomial formula for the quaternions. Takinginto account that H can be considered as a Clifford algebra C`0,n, for n = 2,the generalization of our results for an arbitrary n ≥ 2 will be treated in anextended version of this paper by applying intrinsic properties of monogenicAppell polynomials in terms of several hypercomplex variables.2 Hamilton’s quaternions come into the playConsider a quaternion q ∈ H written asq = x0 + x1i+ x2j+ x3k, where i2 = j2 = k2 = ijk = −1.Due to non-commutativity the formal expansion of a binomial with twoimaginary units (quaternion generators) (i + j)k, k ≥ 0, will not directly lead2to Pascal’s triangle, as the case k = 3 shows:(i+ j)3 = i3 + (iij+ iji+ jii) + (ijj+ jij+ jji) + j3. (4)But that will happen if we try to embed the non-commutative multiplicationinto the concept of a k−nary symmetric (or permutative) operation. Thereforelet ai stay for one of the generators i or j and write the quaternionic k-foldproduct of k − s generators i and s generators j, respectively, in the generalform of a symmetric “× ” product ([13]), i.e.ik−s × js := 1k!∑pi(i1,...,in)ai1ai2 · · · aik (5)where the sum runs over all permutations of all (i1, . . . , in). Then, by takinginto account the repeated use of i and j on the right hand side of (5), we canwriteik−s × js = (k − s)!s!k!∑pi(i1,...,in)ai1ai2 · · · aik =[(ks)]−1 ∑pi(i1,...,in)ai1ai2 · · · aikwhere now the sum runs only over all distinguished permutations of all (i1, . . . , in).Applying, for example, the convention (5) to (4) we obtain now, for k = 3 theexpansion written with binomial coefficients in the form(i+ j)3 =(30)i3 +(31)i2 × j+(32)i× j2 +(33)j3. (6)Analogously, the expansion of (i+ j)k for any k ≥ 0 follows now the rules of theordinary binomial expansion in an evident way and leads to1(i+ j)k =[k∑s=0(ks)(ik−s × js)], k ≥ 0. (7)Needless to say that the generalized binomial formula (7) is a key for studyingcombinatorial relations with quaternions in the following sections.We will show now that another step towards our goal is the evaluation ofexpressions of the form (ik−s × js) k ≥ 0, s = 0, 1, . . . k.3 Evaluating symmetric products of quaterniongeneratorsNotice that for k ≥ 2 the influence of the non-commutativity of the ordinaryquaternionic product is evident. This can be illustrated by the following exam-1An obvious generalization of (5) to the case of more than two generators used in thegeneral case of C`0,n for n ≥ 2 leads to a polynomial formula.3ples:i× j = 1!1!2!(ij+ ij) = 0,i2 × j =(31)−1(iij+ iji+ jii) = −13j, (8)i× j2 =(31)−1(ijj+ jij+ jji) = −13i. (9)To obtain a general rule for those products we refer to an early version ofthe famous Faa´ di Bruno formula for the derivative of a composed function (see[11] and [12]) as it was used in [7].T. Abadie’s formulaIf f and g are real functions of λ, with a sufficient number of derivatives, then(g ◦ f)(s)(λ) =s∑l=0(sl)g(l)(f(λ)){ ds−ldhs−l(∆hf(λ))l}h=0,where ∆hf(λ) :=f(λ+h)−f(λ)h is the difference quotient of f .Consider now the polynomial Fk(λ) of degree k in the real parameter λ,Fk(λ) = (i+ λj)k =k∑s=0(ks)λsik−s × jsand note thatik−s × js = F(s)k (0)s!(ks) . (10)SinceFk(λ) =(−1− λ2)k2 , if k even;(−1− λ2)k−12 (i+ λj), if k odd,(11)it can be composed, for even k, in the form Fk(λ) = (g ◦ f)(λ) with suitablychosen functionsg(λ) = (−1− λ)k2 and f(λ) = λ2.whereas the case of an odd k can be reduced to the previous case by the relationFk(λ) = Fk−1(λ)(i+ λj).Applying T. Abadie’s formula and following the proof of Proposition 1 in [7, p.1730], about generalized powers of hypercomplex variables, one gets finally the4values of (10) in the form:ik−s × js =(−1)k2(k2s2)(ks)−1, k even and s even;0, k even and s odd;(−1)k−12(k−12s2)(ks)−1i, k odd and s even;(−1)k−12(k−12s−12)(ks)−1j, k odd and s odd.(12)Remark 1 The examples in the beginning of this section, in particular relations(8) and (9), confirm very well the last three equalities in (12). It is evident, thatthere are, depending on the relative parities of k resp. s only four types of valuesof ik−s×js, namely (i) real and different from zero, (ii) equal to zero, (iii) a realmultiple of i and (iv) a real multiple of j. The reason for this, at first glance,surprising result is based on the following facts. Obviously, an integer power ofthe type of a so-called reduced purely imaginary quaternion q = i + j, is eithera real number (if k is even) or again a reduced purely imaginary quaternion (ifk is odd; cf. (11)). Besides this, the symmetric products in which such a power(7) is additively decomposed avoid the appearance of ordinary mixed productslike, for example, i·j as mutually annihilating summands. This becomes directlyplausible if we look for example to the binomial expansion for even k = 2 ork = 4 where only entries of type (i) and type (ii) are present. An example foran odd k with 2× k+12 = k+ 1 alternating entries of type (iii) and (iv) is (6). Inboth cases we can recognize the symmetric structure of the corresponding linesin a Pascal triangle with quaternionic entries.4 A combinatorial identity for Vietoris’ numbersequenceBefore coming to our main result, let us still remember a well known combi-natorial identity (cf. [14, p. 130] or [16, p. 44]) that we need for its proof,namelym∑t=0(2tt)(2m− 2tm− t)= 4m, (13)which in turn can be proved by evaluating the square of the generating functionof the central binomial coefficients and its derivatives at x = 0.Now we can proveTheorem 2 Let i and j be two generators of a reduced purely imaginary quater-nion. Then the following combinatorial identity holds(kbk2 c)[ k∑s=0(ks)(ik−s × js)2]= (−2)k. (14)5Taking into account the form of the elements of Vietoris’ number sequence(2), formula (14) can be rewritten in order to obtain the representation of Vi-etoris’ number sequence by symmetric products of the generators i and j.Corollary [Representation of Vietoris’ number sequence]ck = (−1)k[k∑s=0(ks)(ik−s × js)2]−1. (15)Proof of Theorem 2As an auxiliary calculation we determine the square of the symmetric productsin (12) multiplied by(ks)and distinguish between k = 2m and k = 2m+ 1 resp.s = 2t and s = 2t+ 1 for the different parities. We get immediately(ks)(ik−s × js)2 =(mt)2(2m2t) , k = 2m and s = 2t;0, k = 2m and s = 2t+ 1;−(mt )2(2m+12t) , k = 2m+ 1 and s = 2t;−(mt )2(2m+12t+1) , k = 2m+ 1 and s = 2t+ 1.(16)Now we consider two cases corresponding to the parity of k.I. k evenDenote by Ak the left-hand side of (14). The use of (16) (note that thesecond case implies that the sum over odd values of s completely vanishes)together with (13) allows to writeA2m =(2mm)[ 2m∑s=0(2ms)(i2m−s × js)2]=(2mm)[ m∑t=0(2m2t)(i2m−2t × j2t)2 + 0]=(2mm)[ m∑t=0(mt)2(2m2t)] = m∑t=0(2t)!t!t! · (2m−2t)!(m−t)!(m−t)!= 4m = (−2)2mII. k oddIn this case we apply the third and the fourth case of (16) and proceed6analogously to the former case.A2m+1 =(2m+1m)[ 2m+1∑s=0(2ms)(i2m+1−s × js)2]=(2m+1m)[ m∑t=0−(mt )2(2m+12t) − m∑t=0(mt)2(2m+12t+1)]=(2m+1m) m∑t=0(mt)2[ (2t)!(2m−2t+1)!(2m+1)! +(2m−2t)!(2t+1)!(2m+1)!]= −2m∑t=0(2t)!t!t! · (2m−2t)!(m−t)!(m−t)!= −2 · 4m = (−2)2m+1.We finish with examples for the first values of ck in (15) resp. (1),c0 =1c1 =(−1)1[i2 + j2]−1=12c2 =(−1)2[(i2)2 +(21)(i× j)2 + (j2)2]−1=12c3 =(−1)3[(i3)2 +(31)(i2 × j)2 +(32)(i× j2)2 + (j3)2]−1=38c4 =(−1)4[(i4)2 +(41)(i3 × j)2 +(42)(i2 × j2)2 +(43)(i× j3)2 + (j4)2]−1=38.AcknowledgementsThe work of the first and third authors was supported by Portuguese fundsthrough the CIDMA - Center for Research and Development in Mathemat-ics and Applications, and the Portuguese Foundation for Science and Tech-nology (“FCT-Fundac¸a˜o para a Cieˆncia e Tecnologia”), within project PEst-OE/MAT/UI4106/2013. The work of the second author was supported by Por-tuguese funds through the CMAT - Centre of Mathematics and FCT within theProject UID/MAT/00013/2013.References[1] P. Appell, Sur une classe de polynomes, Ann. Sci. E`cole Norm. Sup. 9(2) (1880) 119–144.[2] R. Askey, Orthogonal polynomials and special functions, Society for In-dustrial and Applied Mathematics, Philadelphia, 1975, 2nd printing 1994.7[3] R. Askey and J. 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On Vietoris' number sequence and combinatorial identities with quaternions

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