Generalized Exponentials through Appell sets in Rn+1 and Bessel functions M. I. Falcão∗ and H. R. Malonek† ∗
Universidade do Minho, Portugal
[email protected] † Universidade de Aveiro, Portugal
[email protected]
Abstract. In this paper we present applications of a special class of homogeneous monogenic polynomials constructed, in the framework of hypercomplex function theory, in order to be an Appell set of polynomials. In particular, we derive important properties of an associated exponential function from R3 to R3 and propose a generalization to Rn+1 . Keywords: Appell sets, Bessel functions, hypercomplex function theory PACS: 02.30.-f, 02.30.Gp, 02.30.Lt,02.30.Mv
1. PRELIMINARIES Let {e1 , e2 , · · · , en } be an orthonormal base of the Euclidean vector space Rn with a product according to the multiplication rules ek el + el ek = −2δkl , k, l = 1, · · · , n, where δkl is the Kronecker symbol. This non-commutative product generates the 2n -dimensional Clifford algebra Cl0,n over R and the set {eA : A ⊆ {1, · · · , n}} with eA = eh1 eh2 · · · ehr , 1 ≤ h1 ≤ · · · ≤ hr ≤ n, e0/ = e0 = 1, forms a basis of Cl0,n . The real vector space Rn+1 will be embedded in Cl0,n by identifying the element (x0 , x1 , · · · , xn ) ∈ Rn+1 with the element x = x0 + x of the algebra, where x = ¯ = x02 + x12 + · · · + xn2 . e1 x1 + · · · + en xn .The conjugate of x is x¯ = x0 − x and the norm |x| of x is defined by |x|2 = xx¯ = xx x n n n Denote by ω(x) = |x| ∈ S , where S is the unit sphere in R .
In what follows we consider Cl0,n -valued functions defined in some open subset Ω ⊂ Rn+1 , i.e., functions of the form f (z) = ∑A fA (z)eA , where fA (z) are real valued. We suppose that f is hypercomplex differentiable in Ω in the sense of [1], [2], i.e. has a uniquely defined areolar derivative f 0 in each point of Ω (for the definition of an areolar derivative see [3]). Then f is real differentiable (even real analytic) and f 0 can be expressed in terms of the partial derivatives with respect to xk as f 0 = 1/2(∂0 − ∂x ) f , where ∂0 := ∂∂x , ∂x := e1 ∂∂x + · · · + en ∂∂xn . If now 0 1 D := ∂0 + ∂x is the usual generalized Cauchy-Riemann differential operator, then, obviously f 0 = 1/2D¯ f . Since in [1] it has been shown that a hypercomplex differentiable function belongs to the kernel of D, i.e. satisfies the property D f = 0 ( f is a monogenic function in the sense of Clifford Analysis), then it follows that in fact f 0 = ∂0 f like in the complex case. For our purpose we use monogenic polynomials in terms of the hypercomplex monogenic variables kx zk = xk − x0 ek = − xek +e , k = 1, 2, · · · , n, which leads to generalized powers of degree n that are by convention 2 ν1 symbolically written as z1 ×· · · zνnn and defined as an n-nary symmetric product by zν11 ×· · · zνnn = n!1 ∑π(i1 ,...,in ) zi1 · · · zin , where the sum is taken over all permutations of (i1 , . . . , in ), (see [2] and [3]).
2. THE CASE n = 2 As usual, we identify each element x = (x0 , x1 , x2 ) ∈ R3 with the paravector (also called reduced quaternion) z = x0 + x = x0 + x1 e1 + x2 e2 . In this section, we consider a special set of monogenic basis functions defined and studied to some extend with respect to its algebraic properties in [4], namely functions of the form Pk (x) =
k
∑ Tsk xk−s x¯ s , with Tsk =
s=0
3 1 1 ( 2 )(k−s) ( 2 )(s) , k + 1 (k − s)!s!
(1)
where a(r) denotes the Pochhammer symbol, i.e. a(r) := a(a + 1)(a + 2) · · · (a + r − 1) = and a(0) := 1. In terms of generalized powers these polynomials are of the form n k n k Pk (x) = Pk (z1 , z2 ) = ck ∑ zn−k × z en−k 2 1 1 × e2 , k k=0 where ck :=
k
k!! , (k + 1)!!
if k is odd
s=0
ck−1 ,
if k is even
∑ Tsk (−1)s =
Γ(a+r) Γ(a) ,
for any integer r > 1,
(2)
(3)
In [5], it was proved that the sequence of such polynomials forms a quaternionic Appell sequence. More precisely, these polynomials behave like monomial functions in the sense of the complex powers zk = (x0 + ix1 )k ; k = 1, 2, · · · d k and the power rule of complex differentiation in the form dz z = kzk−1 is extended with respect to the hypercomplex derivative of these special monomial functions, since Pk0 = kPk−1 . Moreover Pk (x) = ck xk
Pk (x0 ) = x0k , and
k
k k−s Pk (x0 + x) = ∑ x0 Ps (x). s=0 s For the above reasons, it is natural to propose the following monogenic function (see [4]), ∞
Exp(x) :=
Pk (x) , k! k=0
∑
(4)
as a generalized exponential function in R3 . This Exp-function, as in the real and complex case, is a solution of the differential equation f 0 = f , f (0) = 1 (5) and satisfies also
Exp0 (λ x) = λ Exp(λ x), λ ∈ R
(6)
Exp(x0 + x) = ex0 Exp(x).
(7)
and Another important property of the real and complex exponential function is that it should be different from zero everywhere. We will prove this property for the exponential function (4) by first writing it in closed form. Theorem 1 The Exp-function (4) can be written in terms of Bessel functions of the first kind, Ja (x), for integer orders a = 0, 1 as Exp(x0 + x) = ex0 (J0 (|x|) + ω(x)J1 (|x|). (8) Proof. We start by first considering the expression of Exp(xi ei ), for i = 1, 2, i.e. ∞
Exp(xi ei ) =
k
∞
s=0
k=0
1
1
k
∑ k! ∑ Tsk (xi ei )k−s (−xi ei )s = ∑ k! ∑ Tsk (−1)s xik eki
k=0
s=0
and applying relation (3) to obtain ∞
Exp(xi ei ) =
∞
∞ 1 1 c2m xi2m (−1)m + ei ∑ c2m+1 xi2m+1 (−1)m . (2m)! (2m + 1)! m=0 m=0
1
∑ k! ck xik eki = ∑
k=0
After some calculation we end up with ∞
Exp(xi ei ) =
∑
m=0
am (−1)m xi2m + ei
xi ∞ ∑ bm (−1)m xi2m , 2 m=0
where am =
(2m − 1)!! 1 = (2m)!(2m)!! 22m (m!)2
Therefore,
and
bm =
2(2m + 1)!! 1 = . (2m + 1)!(2m + 2)!! 22m m!(m + 1)!
xi ∞ (−1)m xi 2m (−1)m xi 2m + e . i ∑ 2 2 m=0 m!(m + 1)! 2 m=0 (m!) 2 ∞
Exp(xi ei ) =
∑
(9)
Recalling the expression of the Bessel functions of the first kind J0 and J1 we can conclude that Exp(xi ei ) = J0 (xi ) + ei J1 (xi ); i = 1, 2.
(10)
We note that it is possible to express Exp(x) almost in the same way. In fact, let x = x1 e1 + x2 e2 =
x1 e1 + x2 e2 |x| = ω(x)|x|. |x|
Since ω(x)2 = −1, then it follows from (9) and (10): ∞
Exp(x) =
(−1)m |x| 2m |x| ∞ (−1)m |x| 2m + ω(x) = J0 (|x|) + ω(x)J1 (|x|). ∑ 2 2 m=0 m!(m + 1)! 2 m=0 (m!) 2
∑
The final result follows at once from (7). Straightforward calculations lead to the following important properties of the exponential function (4). Corollary 1 1. Exp(x) has no zeros. 2. If J(Exp(x)) denotes the jacobian of the function Exp(x), then 3x0 − e|x|2 J1 (|x|) J0 (|x|)J2 (|x|)|x| − J0 (|x|)J1 (|x|) − J1 (|x|)2 |x| , J(Exp(x0 + x)) = 1 e3x0 , 4
if x 6= 0 if x = 0
We underline that the zeros of the jacobian of Exp(x) are exactly the zeros of J1 (|x|) and therefore they are all laid on concentric cylinders with radius corresponding to de zeros of J1 (|x|). In particular, the first positive zero is ζ1 ≈ 3.831705970. Thus, we can conclude, for example, that on the cylinder C := {(x0 , x1 , x2 ) ∈ R3 : x12 +x22 < ζ12 }, the Exp-function can be treated as a mapping function from R3 to R3 . Images of some domains in R3 by this exponential function can be found in [4].
3. A GENERALIZED EXPONENTIAL FUNCTION FROM Rn+1 TO Rn+1 In the case x = x0 + x = x0 + x1 e1 + x2 e2 + · · · + xn en , we denote by Pkn (x) the homogeneous monogenic polynomial of degree k, generalizing the particular n = 2 case in (1). It can be proved that for arbitrary n ≥ 1, the polynomials Pkn (x) =
k
∑ Tsk (n) xk−s x¯s , with Tsk (n) =
s=0
n+1 n−1 n! ( 2 )(k−s) ( 2 )(s) , (n)k (k − s)!s!
form an Appell sequence. In terms of generalized powers these polynomials are of the form n ν1 Pkn (x) = Pk (z1 , · · · , zn ) = ck (n) ∑ zν11 × · · · × zνnn e1 × · · · × eνnn , ν |ν|=n
(11)
(12)
where ν = (ν1 , · · · , νn ) is a multiindex and
ck (n) :=
1 k! n + 1 , if k is odd k−1 n 2 k−1 (k) ( 2 ) ( 2 )!
∑ks=0 (−1)s Tsk (n) = k! n + 1 ∑ks=0 Tsk (n) 1 , n k 2 k ( (k) ( ) 2 )!
(13)
if k is even
2
Now if we define
∞
Expn (x) :=
Pkn (x) k! k=0
∑
(14)
then we can prove the following result. Theorem 2 The Expn -function (14) can be written in terms of Bessel functions of the first kind, Ja (x), for orders a = 2n − 1, n2 as n −1 n 2 2 Expn (x0 + x) = ex0 Γ( ) J n2 −1 (|x|) + ω(x)J n2 (|x|) . (15) 2 |x| For n = 1 and n = 2, (15) gives the ordinary complex exponential and the reduced quaternion valued Exp-function (4), respectively. Moreover, it is easy to conclude that this function is different from zero everywhere and fulfills properties (5)-(7), for any arbitrary n ≥ 1. For the particular case n = 3, (15) leads to sin(|x|) − |x| cos(|x|) x0 sin(|x|) + ω(x) . (16) Exp3 (x0 + x) = e |x| |x| This Exp3 -function coincides with the one referred by Sprössig in [6], based on results of [7, 8, 9, 10] which are generalizations of Fueter’s approach. We would like to point out that the homogeneous monogenic polynomials (1) constructed so that they form an Appell sequence, are particularly easy to handle and can play an important role in 3D-mapping problems. For a case study of such approach, see for example [11].
ACKNOWLEDGMENTS The research of the first author was partially supported by the Portuguese Foundation for Science and Technology (FCT) through the research program POCTI. The research of the second author was partially supported by the R&D unit Matemática e Aplicações (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT). The second author thanks N. Gürlebeck for fruitful discussions on this subject.
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