The so-called Vietoris' number sequence is a sequence of rational numbers that appeared for the first time in a celebrated theorem by Vietoris (1958) about the positivity of certain trigonometric sums with important applications in harmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphic functions (Ruscheweyh/ Salinas, 2004). In the context of hypercomplex function theory those numbers appear as coefficients of special homogeneous polynomials in R^3 whose generalization to an arbitrary dimension n lead to a n-parameter generalized Vietoris' number sequence that characterizes hypercomplex Appell polynomials in R^n.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. The work of the other 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 PEst-OE/MAT/UI4106/2013

Cação, Isabel

Falcão, M. I.

Malonek, Helmuth

English

The so-called Vietoris' number sequence is a sequence of rational numbers
that appeared for the  rst time in a celebrated theorem by Vietoris (1958)
about the positivity of certain trigonometric sums with important applications in
harmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphic
functions (Ruscheweyh/ Salinas, 2004). In the context of hypercomplex function
theory those numbers appear as coe cients of special homogeneous polynomials
in R3 whose generalization to an arbitrary dimension n lead to a n-parameter
generalized Vietoris' number sequence that characterizes hypercomplex Appell
polynomials in Rn.publishe

Repositório Institucional da Universidade de Aveiro

PROCEEDINGS BOOK OF MICOPAM 2018Vietoris’ number sequence and itsgeneralizations through hypercomplexfunction theoryI. Cac¸a˜o1, M. I. Falca˜o2, H. R. Malonek1AbstractThe so-called Vietoris’ number sequence is a sequence of rational numbersthat appeared for the first time in a celebrated theorem by Vietoris (1958)about the positivity of certain trigonometric sums with important applications inharmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphicfunctions (Ruscheweyh/ Salinas, 2004). In the context of hypercomplex functiontheory those numbers appear as coefficients of special homogeneous polynomialsin R3 whose generalization to an arbitrary dimension n lead to a n-parametergeneralized Vietoris’ number sequence that characterizes hypercomplex Appellpolynomials in Rn.2010 Mathematics Subject Classifications: 30G35, 11B83, 05A10Keywords: Vietoris’ number sequence, monogenic Appell polynomials, generat-ing functionsIntroductionThe Vietoris’ number sequence S is the following 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 be writtenin compact form (cf. [3]) as S = (ck)k≥0, whereck =12k(kbk2 c)=(12)bk+12 c(1)bk+12 c. (2)Here, as usual, b ·c denotes the floor function and (·)k is the raising factorial in theclassical form of the Pochhammer symbol.Seemingly this sequence appeared, for the first time, in the context of positivetrigonometric sums in a celebrated paper of L. Vietoris [11]. 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,wherea2k = a2k+1 =( 12 )kk!, k = 0, 1, . . . . (3)Dedicated to Professor G. Milovanovic´ 162 Antalya-TURKEYPROCEEDINGS BOOK OF MICOPAM 2018We call attention to the fact that because of (3), the coefficients in the sine sumare exactly the elements of S in (2) or, explicitly, in (1). Compared with the tra-ditional way of defining the coefficient sequence by (3), the use of the properties ofthe generalized central binomial coefficient allows a unique representation (2) withconsecutively running index k.In the context of hypercomplex function theory, the sequence S characterizes aspecial homogeneous polynomial sequence that can be considered in higher dimensionsas the counterpart of the sequence of holomorphic powers.In the sequel we will use the following basic concepts and notations. Let {e1, e2, . . . , en}be an orthonormal basis of the Euclidean vector space Rn endowed with a non-commutative product according to the multiplication ruleseiej + ejei = −2δij , i, j = 1, 2, . . . , n,where δij is the Kronecker symbol. This generates the associative 2n-dimensionalClifford algebra C`0,n over R, whose elements are of the form a =∑A aAeA, aA ∈ R,with A ⊆ {1, · · · , n}, eA = el1el2 · · · elr , where 1 ≤ l1 < · · · < lr ≤ n and e∅ = e0 = 1.In general, the vector space Rn+1 is embedded in C`0,n by identifying the element(x0, x1, . . . , xn) ∈ Rn+1 with the element (paravector)x = x0 +n∑k=1ekxk = x0 + x ∈ An := spanR{1, e1, . . . , en} ⊂ C`0,n.Its conjugate is x¯ = x0 − x and the norm of x is given by |x| = (xx¯)1/2 = (x¯x)1/2 =(∑nk=0 x2k)1/2.We consider C`0,n-valued functions defined as mappingsf : Ω ⊂ Rn+1 ∼= An 7−→ C`0,nsuch that f(x) =∑A fA(x)eA, fA(x) ∈ R and Ω is an open subset of Rn+1, n ≥ 1.The generalized Cauchy-Riemann operator in Rn+1 is ∂ := 12 (∂0 +∂x), with ∂0 :=∂∂x0, and ∂x :=∑nk=1 ek∂∂xk. A C1-function f is called left (right) monogenic,or simply monogenic in Rn+1 if it is a solution of the differential equation ∂f = 0(f∂ = 0).Notice that the operator ∂ := 12 (∂0 − ∂x) is the conjugate generalized Cauchy-Riemann operator and acts as derivative of a monogenic function (cf.[9]). Thereforethe hypercomplex derivative of a monogenic function f can be calculated as ∂f =12 (∂0 − ∂x)f = ∂0f , i.e. in the same way as the complex derivative of a holomorphicfunction.Main ResultsIn the center of our attention is the sequence of paravector-valued monogenicpolynomials (Pnk )n∈N such that∂Pnk (x) = kPnk−1(x), x ∈ An, k = 1, 2, . . . . (4)Choosing as initial value Pn0 = 1, the recurrence (4) together with the requirement ofmonogeinity,∂Pnk (x) = 0, x ∈ An,lead to the explicit representationPnk (x) =k∑s=0(ks)cs(n)xk−s0 xs, x ∈ An, (5)Dedicated to Professor G. Milovanovic´ 163 Antalya-TURKEYPROCEEDINGS BOOK OF MICOPAM 2018wherecs(n) :=( 12 )b k+12 c(n2 )b k+12 c, s = 0, . . . , k, k = 1, 2, . . . . (6)See [4, 7, 8] for details.The equality (4) for monogenic homogeneous polynomials generalizes to higherdimensions the classical concept of Appell polynomials (cf.[1]). We remark that theinitial value of a Clifford algebra-valued Appell polynomial sequence can be a real, aClifford number or a vector-valued monogenic polynomial of a fixed degree (a mono-genic constant) (see [6, 10]).Taking into account that for n = 2, (6) gives exactly the rational numbers (2)that constitute the Vietoris’ number sequence S, the coefficients sequence (S(n))n∈Ncharacterizing the Appell polynomials (5) and whose general term is given by (6) isa n-parameter generalization of S.Moreover, the representation (6) in terms of quotients of numbers represented bythe Pochhammer symbol suggests the use of the well known Gauss’ hypergeometricfunction2F1(a, b; c; z) =+∞∑k=0(a)k (b)k(c)kzkk!, |z| < 1, a, b ∈ C, c ∈ C \ Z−0to derive a generating function of the generalized Vietoris’ sequence (S(n))n∈N. Infact, in [5] the following result was obtained.Theorem 2. Let G(., n) be the following real-valued function depending on a param-eter n ∈ N:G(t;n) ={1t[(1 + t) 2F1(12 , 1;n2 ; t2)− 1] , if t ∈]− 1, 0[∪]0, 1[1, if t = 0.Then, for any fixed n ∈ N, G(., n) is a one-parameter generating function of thesequence S(n).It is clear that we can obtain a closed formula for the generating function of thesequence (S(n))n∈N as long as a closed formula for the corresponding hypergeometricseries is known. As examples we list some cases where closed formulae can be easilyobtained:1. n = 1In this case, ck(1) = 1 (k ≥ 0) and the corresponding generating function isgiven byG(t; 1) = 1t[(1 + t) 2F1(12 , 1;12 ; t2)− 1] = 11−t ,because 2F1(12 , 1;12 ; t2) reduces to the geometric function.2. n = 2Recalling (2), we have c2k(2) = c2k−1(2) =( 12 )kk! andG(t; 2) = 1t[(1 + t) 2F1(12 , 1; 1; t2)− 1] = √1+t−√1−tt√1−t .3. n = 3The generalized Vietoris’ numbers are c2k(3) = c2k−1(3) = 12k+1 and the corre-sponding generating function is given byG(t; 3) = 1t[(1 + t) 2F1(12 , 1;32 ; t2)− 1] = 1t ( t+1t ln√ 1+t1−t − 1) .Dedicated to Professor G. Milovanovic´ 164 Antalya-TURKEYPROCEEDINGS BOOK OF MICOPAM 20184. n = 4In this case, c2k(4) = c2k−1(4) =( 12 )k(k+1)! andG(t; 4) = 1t[(1 + t) 2F1(12 , 1; 2; t2)− 1] = 2t+1−√1−t2t(1+√1−t2) .ConclusionBy providing a link between the Vietoris’ number sequence and hypercomplex Ap-pell polynomials, we were able to define one-parameter generalized Vietoris’ numbersequences and obtain their generating functions.AcknowledgementsThe work of the second author was supported by Portuguese funds through theCMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.The work of the other authors was supported by Portuguese funds through theCIDMA - Center for Research and Development in Mathematics and Applications,and the Portuguese Foundation for Science and Technology (“FCT-Fundac¸a˜o para aCieˆncia e Tecnologia”), within project PEst-OE/MAT/UI4106/2013.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 Industrialand Applied Mathematics, Philadelphia, 2nd ed. 1994.[3] I. Cac¸a˜o, M. I. Falca˜o, and H. R. Malonek, Matrix representations of a basicpolynomial sequence in arbitrary dimension, Comput. Methods Funct. Theory 12(2) (2012) 371–391.[4] I. Cac¸a˜o, M. I. Falca˜o, and H. R. Malonek, Hypercomplex Polynomials, Vietoris’Rational Numbers and a Related Integer Numbers Sequence, Complex Anal. Oper.Theory 11 (5) (2017) 1059-1076.[5] I. Cac¸a˜o, M. I. Falca˜o, and H. R. Malonek, On generalized Vietoris’ numbersequences, to appear.[6] D. Pen˜a Pen˜a: Shifted Appell Sequences in Clifford Analysis. Results. Math. 63,1145–1157 (2013)[7] M. I. Falca˜o, H. R. Malonek, Generalized exponentials through Appell sets inRn+1 and Bessel functions. In: T. E. Simos, G. Psihoyios, C. Tsitouras (Eds.),AIP Conference Proceedings 936, 2007, 738–741.[8] M. I. Falca˜o, H. R. Malonek, A note on a one-parameter family of non-symmetricnumber triangles, Opuscula Mathematica 32 (4) (2012), 661–673.[9] K. Gu¨rlebeck, H. R. Malonek: A hypercomplex derivative of monogenic functionsin Rm+1 and its applications. Complex Variables 39, 199-228 (1999)Dedicated to Professor G. Milovanovic´ 165 Antalya-TURKEYPROCEEDINGS BOOK OF MICOPAM 2018[10] R. La`vicˇka: Complete Orthogonal Appell Systems for Spherical Monogenics.Complex Anal. Oper. Theory, 6, 477–489 (2012)[11] L. Vietoris, U¨ber das Vorzeichen gewisser trigonometrischer Summen, Sitzungs-ber. O¨sterr. Akad. Wiss 167, (1958) 125–135.1CIDMA, University of Aveiro, Portugal2CMAT and Department of Mathematics and Applications, Univer-sity of Minho, PortugalE-mail : isabel.cacao@ua.pt, mif@math.uminho.pt, hrmalon@ua.ptDedicated to Professor G. Milovanovic´ 166 Antalya-TURKEY

Vietoris' number sequence and its generalizations through hypercomplex function theory

Universidade do Minho: RepositoriUM

Malonek, Helmuth R.

PROCEEDINGS BOOK OF MICOPAM 2018Vietoris’ number sequence and itsgeneralizations through hypercomplexfunction theoryI. Cação1, M. I. Falcão2, H. R. Malonek1AbstractThe so-called Vietoris’ number sequence is a sequence of rational numbersthat appeared for the first time in a celebrated theorem by Vietoris (1958)about the positivity of certain trigonometric sums with important applications inharmonic analysis (Askey/Steinig, 1974) and in the theory of stable holomorphicfunctions (Ruscheweyh/ Salinas, 2004). In the context of hypercomplex functiontheory those numbers appear as coefficients of special homogeneous polynomialsin R3 whose generalization to an arbitrary dimension n lead to a n-parametergeneralized Vietoris’ number sequence that characterizes hypercomplex Appellpolynomials in Rn.2010 Mathematics Subject Classifications: 30G35, 11B83, 05A10Keywords: Vietoris’ number sequence, monogenic Appell polynomials, generat-ing functionsIntroductionThe Vietoris’ number sequence S is the following 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 be writtenin compact form (cf. [3]) as S = (ck)k≥0, whereck =12k(kbk2 c)=(12)bk+12 c(1)bk+12 c. (2)Here, as usual, b ·c denotes the floor function and (·)k is the raising factorial in theclassical form of the Pochhammer symbol.Seemingly this sequence appeared, for the first time, in the context of positivetrigonometric sums in a celebrated paper of L. Vietoris [11]. Askey’s version [2, p. 5]of Vietoris’ theorem is the following:Theorem 1 (L. Vietoris).n∑k=1ak sin kθ > 0, 0 < θ < π, andn∑k=0ak cos kθ > 0, 0 ≤ θ < π,wherea2k = a2k+1 =( 12 )kk!, k = 0, 1, . . . . (3)Dedicated to Professor G. Milovanović 162 Antalya-TURKEYPROCEEDINGS BOOK OF MICOPAM 2018We call attention to the fact that because of (3), the coefficients in the sine sumare exactly the elements of S in (2) or, explicitly, in (1). Compared with the tra-ditional way of defining the coefficient sequence by (3), the use of the properties ofthe generalized central binomial coefficient allows a unique representation (2) withconsecutively running index k.In the context of hypercomplex function theory, the sequence S characterizes aspecial homogeneous polynomial sequence that can be considered in higher dimensionsas the counterpart of the sequence of holomorphic powers.In the sequel we will use the following basic concepts and notations. Let {e1, e2, . . . , en}be an orthonormal basis of the Euclidean vector space Rn endowed with a non-commutative product according to the multiplication ruleseiej + ejei = −2δij , i, j = 1, 2, . . . , n,where δij is the Kronecker symbol. This generates the associative 2n-dimensionalClifford algebra C`0,n over R, whose elements are of the form a =∑A aAeA, aA ∈ R,with A ⊆ {1, · · · , n}, eA = el1el2 · · · elr , where 1 ≤ l1 < · · · < lr ≤ n and e∅ = e0 = 1.In general, the vector space Rn+1 is embedded in C`0,n by identifying the element(x0, x1, . . . , xn) ∈ Rn+1 with the element (paravector)x = x0 +n∑k=1ekxk = x0 + x ∈ An := spanR{1, e1, . . . , en} ⊂ C`0,n.Its conjugate is x̄ = x0 − x and the norm of x is given by |x| = (xx̄)1/2 = (x̄x)1/2 =(∑nk=0 x2k)1/2.We consider C`0,n-valued functions defined as mappingsf : Ω ⊂ Rn+1 ∼= An 7−→ C`0,nsuch that f(x) =∑A fA(x)eA, fA(x) ∈ R and Ω is an open subset of Rn+1, n ≥ 1.The generalized Cauchy-Riemann operator in Rn+1 is ∂ := 12 (∂0 +∂x), with ∂0 :=∂∂x0, and ∂x :=∑nk=1 ek∂∂xk. A C1-function f is called left (right) monogenic,or simply monogenic in Rn+1 if it is a solution of the differential equation ∂f = 0(f∂ = 0).Notice that the operator ∂ := 12 (∂0 − ∂x) is the conjugate generalized Cauchy-Riemann operator and acts as derivative of a monogenic function (cf.[9]). Thereforethe hypercomplex derivative of a monogenic function f can be calculated as ∂f =12 (∂0 − ∂x)f = ∂0f , i.e. in the same way as the complex derivative of a holomorphicfunction.Main ResultsIn the center of our attention is the sequence of paravector-valued monogenicpolynomials (Pnk )n∈N such that∂Pnk (x) = kPnk−1(x), x ∈ An, k = 1, 2, . . . . (4)Choosing as initial value Pn0 = 1, the recurrence (4) together with the requirement ofmonogeinity,∂Pnk (x) = 0, x ∈ An,lead to the explicit representationPnk (x) =k∑s=0(ks)cs(n)xk−s0 xs, x ∈ An, (5)Dedicated to Professor G. Milovanović 163 Antalya-TURKEYPROCEEDINGS BOOK OF MICOPAM 2018wherecs(n) :=( 12 )b k+12 c(n2 )b k+12 c, s = 0, . . . , k, k = 1, 2, . . . . (6)See [4, 7, 8] for details.The equality (4) for monogenic homogeneous polynomials generalizes to higherdimensions the classical concept of Appell polynomials (cf.[1]). We remark that theinitial value of a Clifford algebra-valued Appell polynomial sequence can be a real, aClifford number or a vector-valued monogenic polynomial of a fixed degree (a mono-genic constant) (see [6, 10]).Taking into account that for n = 2, (6) gives exactly the rational numbers (2)that constitute the Vietoris’ number sequence S, the coefficients sequence (S(n))n∈Ncharacterizing the Appell polynomials (5) and whose general term is given by (6) isa n-parameter generalization of S.Moreover, the representation (6) in terms of quotients of numbers represented bythe Pochhammer symbol suggests the use of the well known Gauss’ hypergeometricfunction2F1(a, b; c; z) =+∞∑k=0(a)k (b)k(c)kzkk!, |z| < 1, a, b ∈ C, c ∈ C \ Z−0to derive a generating function of the generalized Vietoris’ sequence (S(n))n∈N. Infact, in [5] the following result was obtained.Theorem 2. Let G(., n) be the following real-valued function depending on a param-eter n ∈ N:G(t;n) ={1t[(1 + t) 2F1(12 , 1;n2 ; t2)− 1], if t ∈]− 1, 0[∪]0, 1[1, if t = 0.Then, for any fixed n ∈ N, G(., n) is a one-parameter generating function of thesequence S(n).It is clear that we can obtain a closed formula for the generating function of thesequence (S(n))n∈N as long as a closed formula for the corresponding hypergeometricseries is known. As examples we list some cases where closed formulae can be easilyobtained:1. n = 1In this case, ck(1) = 1 (k ≥ 0) and the corresponding generating function isgiven byG(t; 1) = 1t[(1 + t) 2F1(12 , 1;12 ; t2)− 1]= 11−t ,because 2F1(12 , 1;12 ; t2) reduces to the geometric function.2. n = 2Recalling (2), we have c2k(2) = c2k−1(2) =( 12 )kk! andG(t; 2) = 1t[(1 + t) 2F1(12 , 1; 1; t2)− 1]=√1+t−√1−tt√1−t .3. n = 3The generalized Vietoris’ numbers are c2k(3) = c2k−1(3) =12k+1 and the corre-sponding generating function is given byG(t; 3) = 1t[(1 + t) 2F1(12 , 1;32 ; t2)− 1]= 1t(t+1t ln√1+t1−t − 1).Dedicated to Professor G. Milovanović 164 Antalya-TURKEYPROCEEDINGS BOOK OF MICOPAM 20184. n = 4In this case, c2k(4) = c2k−1(4) =( 12 )k(k+1)! andG(t; 4) = 1t[(1 + t) 2F1(12 , 1; 2; t2)− 1]= 2t+1−√1−t2t(1+√1−t2) .ConclusionBy providing a link between the Vietoris’ number sequence and hypercomplex Ap-pell polynomials, we were able to define one-parameter generalized Vietoris’ numbersequences and obtain their generating functions.AcknowledgementsThe work of the second author was supported by Portuguese funds through theCMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.The work of the other authors was supported by Portuguese funds through theCIDMA - Center for Research and Development in Mathematics and Applications,and the Portuguese Foundation for Science and Technology (“FCT-Fundação para aCiência e Tecnologia”), within project PEst-OE/MAT/UI4106/2013.References[1] P. Appell, Sur une classe de polynomes, Ann. Sci. Ècole Norm. Sup. 9 (2) (1880)119–144.[2] R. Askey, Orthogonal polynomials and special functions, Society for Industrialand Applied Mathematics, Philadelphia, 2nd ed. 1994.[3] I. Cação, M. I. Falcão, and H. R. Malonek, Matrix representations of a basicpolynomial sequence in arbitrary dimension, Comput. Methods Funct. Theory 12(2) (2012) 371–391.[4] I. Cação, M. I. Falcão, and H. R. Malonek, Hypercomplex Polynomials, Vietoris’Rational Numbers and a Related Integer Numbers Sequence, Complex Anal. Oper.Theory 11 (5) (2017) 1059-1076.[5] I. Cação, M. I. Falcão, and H. R. Malonek, On generalized Vietoris’ numbersequences, to appear.[6] D. Peña Peña: Shifted Appell Sequences in Clifford Analysis. Results. Math. 63,1145–1157 (2013)[7] M. I. Falcão, H. R. Malonek, Generalized exponentials through Appell sets inRn+1 and Bessel functions. In: T. E. Simos, G. Psihoyios, C. Tsitouras (Eds.),AIP Conference Proceedings 936, 2007, 738–741.[8] M. I. Falcão, H. R. Malonek, A note on a one-parameter family of non-symmetricnumber triangles, Opuscula Mathematica 32 (4) (2012), 661–673.[9] K. Gürlebeck, H. R. Malonek: A hypercomplex derivative of monogenic functionsin Rm+1 and its applications. Complex Variables 39, 199-228 (1999)Dedicated to Professor G. Milovanović 165 Antalya-TURKEYPROCEEDINGS BOOK OF MICOPAM 2018[10] R. Làvička: Complete Orthogonal Appell Systems for Spherical Monogenics.Complex Anal. Oper. Theory, 6, 477–489 (2012)[11] L. Vietoris, Über das Vorzeichen gewisser trigonometrischer Summen, Sitzungs-ber. Österr. Akad. Wiss 167, (1958) 125–135.1CIDMA, University of Aveiro, Portugal2CMAT and Department of Mathematics and Applications, Univer-sity of Minho, PortugalE-mail : isabel.cacao@ua.pt, mif@math.uminho.pt, hrmalon@ua.ptDedicated to Professor G. Milovanović 166 Antalya-TURKEY

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Vietoris' number sequence and its generalizations through hypercomplex function theory

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