Generating functions for spherical harmonics and spherical monogenics

arXiv:1404.4066v1 [math.CV] 15 Apr 2014

Generating functions for spherical harmonics and spherical monogenics P. Cerejeiras, U. K¨ahler and R. L´aviˇcka To K. G¨ urlebeck

Abstract. In this paper, we study generating functions for the standard orthogonal bases of spherical harmonics and spherical monogenics in Rm . Here spherical monogenics are polynomial solutions of the Dirac equation in Rm . In particular, we obtain the recurrence formula which expresses the generating function in dimension m in terms of that in dimension m−1. Hence we can find closed formulæ of generating functions in Rm by induction on the dimension m. Mathematics Subject Classification (2010). 30G35, 33C55, 33C45. Keywords. spherical harmonics, spherical monogenics, Gelfand-Tsetlin basis, orthogonal basis, generating function.

1. Introduction It is well-known that classical orthogonal polynomials can be defined by their generating functions. For example, the Gegenbauer polynomials Ckν are uniquely determined by the generating function ∞ X 1 = Ckν (x)hk (1) (1 − 2xh + h2 )ν k=0

where ν > 0, |x| ≤ 1 and |h| < 1 (see e.g. [14, p. 18] or [24, p.173]). In [24], a general framework is developed for a study of properties of polynomial sequences, including the Appell property and generating functions. In this paper, we deal with generating functions for the standard orthogonal bases of spherical harmonics and spherical monogenics in Rm . Orthogonal bases of spherical harmonics are well-known and have been studied for a long time. Spherical harmonics are useful in many theoretical areas and on applications such as structural mechanics, etc. In Clifford analysis, a similar role is played by spherical monogenics. Monogenic functions are defined as Clifford algebra valued solutions f of the equation ∂f = 0 where ∂

2

P. Cerejeiras, U. K¨ ahler and R. L´ aviˇcka

is the Dirac operator on Rm . Spherical monogenics are polynomial solutions of the Dirac equation. Since the Dirac operator ∂ factorizes the Laplace operator ∆ in the sense that ∆ = −∂ 2 Clifford analysis can be understood as a refinement of harmonic analysis. On the other hand, monogenic functions are at the same time a higher dimensional analogue of holomorphic functions of one complex variable. See [3, 13, 17, 16] for an account of Clifford analysis. The first construction of orthogonal bases of spherical monogenics valid for any dimension was given by F. Sommen, see [25, 13]. In dimension 3, explicit constructions using the standard bases of spherical harmonics were done also by K. G¨ urlebeck, H. Malonek, I. Ca¸ca˜o and S. Bock (see e.g. [1, 6, 7, 8, 9, 10, 11]). From the point of view of representation theory, the standard bases of spherical harmonics are nothing else than examples of the so-called Gelfand-Tsetlin bases, see [23]. V. Souˇcek proposed studying these bases in Clifford analysis. In particular, in [2], it is observed that the complete orthogonal system in R3 of [1] and F. Sommen’s bases [25, 13] can be both considered as Gelfand-Tsetlin bases. Actually, it turns out that Gelfand-Tsetlin bases in all cases so far studied in Clifford analysis are, by construction, uniquely determined and orthogonal and, in addition, they possess the so-called Appell property, see [22] for a recent survey, [19, 20] for the classical Clifford analysis, [12, 21] for Hodge-de Rham systems and [4, 5] for Hermitian Clifford analysis. Therefore we call them the standard orthogonal bases in the sequel. For a detailed historical account of this topic, we refer to [2]. In this paper, we study generating functions for the standard orthogonal bases of spherical harmonics and spherical monogenics in Rm . We obtain the recurrence formula which expresses the generating function in dimension m in terms of that in dimension m−1, see below Theorem 1 for spherical harmonics and Theorem 2 for spherical monogenics. Using the recurrence formula, we can obtain closed formulæ of generating functions in Rm by induction on the dimension m. This is based on the generating function (1) for the Gegenbauer polynomials. It seems that analogous results can be obtained also for Hodgede Rham systems [21] and even in Hermitian Clifford analysis [5]. But, in the hermitian case, the generating function for the Jacobi polynomials should be used instead of (1).

2. Spherical harmonics In this section, we study generating functions for spherical harmonics. Let us recall the standard construction of an orthogonal basis in the complex Hilbert space L2 (Bm , C) ∩ Ker ∆ of L2 -integrable harmonic functions g : Bm → C. Here Bm is the unit ball in Rm . One proceeds by induction. Of course, the polynomials k2 harm± k2 (x1 , x2 ) = (x1 ± ix2 ) /(k2 !), k2 ∈ N0

(2)

Generating functions

3

form an orthogonal basis of the space L2 (B2 , C) ∩ Ker ∆. To construct the (k ) (k ) bases in higher dimensions, we need the embedding factors Fm,jm = Fm,jm (x) defined as m/2+j−1 (k ) (xm /|x|m ), x ∈ Rm (3) Fm,jm = |x|kmm Ckm p 2 where x = (x1 , . . . , xm ) and |x|m = x1 + · · · + x2m . Then, it is well-known that an orthogonal basis of the space L2 (Bm , C) ∩ Ker ∆ is formed by the polynomials m Y (k ) ± Fr,kr∗ (4) (x , x ) harm± (x) = harm 1 2 k k2 r−1

r=3

where k = (k2 , · · · , km ) ∈ Nm−1 and kr∗ = k2 + · · · + kr . See e.g. [14, p. 35] 0 or [20]. In difference to [20], we use another normalization of the embedding (k ) factors Fm,jm and we also change the notation for indices which in turns provides a more elegant expression for generating functions. ± Definition 1. We define the generating function Hm of the orthogonal basis ± m−1 m harmk , k ∈ N0 of spherical harmonics in R by X ± k Hm (x, h) = harm± k (x) h k∈Nm−1 0

whenever the series on the right-hand side converges absolutely. Here x ∈ Rm , h = (h2 , . . . , hm ) ∈ Rm−1 and hk = hk22 · · · hkmm . Obviously, the following result follows easily from (1). Lemma 1. We have that ∞ X (k ) Fm,jm (x) hkmm = km =0

1 m (1 − 2xm hm + h2m |x|2m ) 2 −1+j

where |x|m ≤ 1, |hm | < 1 and j ∈ N0 . ± Now we prove basic properties of the generating functions Hm .

Theorem 1. For each m ≥ 2 there is a neighborhood Um of 0 in Rm−1 such that the following statements hold true. ± (x, h) are defined if |x|m ≤ 1 and h ∈ Um . (i) The generating functions Hm m−1 (ii) For each k ∈ N0 , we have that

harm± k (x) =

1 k ± ∂ Hm (x, h)|h=0 , |x|m ≤ 1 k!

m where k! = (k2 !) · · · (km !) and ∂ k = ∂hk22 · · · ∂hkm . (iii) For m ≥ 3, |x|m ≤ 1 and h ∈ Um , we have that

1− m 2

± Hm (x, h) = dm

± Hm−1 (x, h/dm )

where dm = 1 − 2xm hm + h2m |x|2m , x = (x1 , · · · , xm−1 ) and h/dm = (h2 /dm , · · · , hm−1 /dm ).

4

P. Cerejeiras, U. K¨ ahler and R. L´ aviˇcka

Proof. We prove this theorem by induction on the dimension m. It is easily seen that the theorem is true for m = 2. Indeed, we have that H2± (x1 , x2 , h2 ) =

∞ X (x1 ± ix2 )k2 k2 h2 = exp((x1 ± ix2 )h2 ). k2 !

k2 =0

± Now assume that the theorem is true for m − 1. Let Hm−1 (x, h) be defined for h ∈ Um−1 = (−δ2 , δ2 ) × · · · × (−δm−1 , δm−1 ) and |x|m−1 ≤ 1 and let |x|m ≤ 1. It is easy to see that ! ∞ X X (km ) k km ± Fm,k∗ (x) hm harm± Hm (x, h) = (5) k (x) h m−1

km =0

k

. By Lemma where the first sum is taken over all k = (k2 , · · · , km−1 ) ∈ Nm−2 0 1, we have that ∞ X

1− m 2 −(k2 +···+km−1 )

(k )

Fm,km∗

m−1

(x) hkmm = dm

km =0

if |hm | < 1. Using this formula and (5), we have that X 1− m 1− m ± ± k 2 Hm (x, h) = dm 2 harm± Hm−1 (x, h/dm ) k (x) (h/dm ) = dm k

whenever h ∈ Um = (−δ2 /4, δ2 /4) × · · · × (−δm−1 /4, δm−1/4) × (−1/2, 1/2). Indeed, dm ≥ (1 − hm |x|m )2 > 1/4 if |hm | < 1/2. Hence, if h ∈ Um we have that h/dm ∈ Um−1 and, by (5), we can easily see that some rearrangement of ± the power series defining Hm (x, h) converges at h. Then Abel’s Lemma [18, Proposition 1.5.5, p. 23] proves that this power series converges absolutely on the whole Um , which finishes the proof of the theorem.  Using the recurrence formula (iii) of Theorem 1, we can find closed formulæ of generating functions for spherical harmonics in Rm by induction on the dimension m. Corollary 1. In particular, we have the following formula   1 (x1 ± ix2 )h2 H3± (x1 , x2 , x3 , h2 , h3 ) = . exp 1 − 2x3 h3 + h23 |x|23 (1 − 2x3 h3 + h23 |x|23 )1/2 Remark 1. It is well-known that an orthogonal basis of real valued spherical + m−1 harmonics in Rm is formed by the polynomials ℜharm+ . k , ℑharmk , k ∈ N0 Here ℜz and ℑz are the real and imaginary part of the complex number z. + + . Hence the corresponding generating functions are ℜHm , ℑHm Remark 2. If one replaces in the definition of the orthogonal basis (4) the k2 polynomials harm± k2 (x1 , x2 ) = (x1 ± ix2 ) /(k2 !) with ±

harmk2 (x1 , x2 ) = (x1 ± ix2 )k2 ,

(6)

Generating functions

5

±

the corresponding generating functions H m are definitely different from Hm± but they obviously satisfy again Theorem 1. In particular, we have that ±

H 2 (x1 , x2 , h2 ) =

∞ X

(x1 ± ix2 )k2 hk22 =

k2 =0

1 − (x1 ∓ x2 i)h2 . 1 − 2x1 h2 + h22 |x|22

3. Spherical monogenics In this section, we introduce and investigate generating functions for spherical monogenics. For an account of Clifford analysis, we refer to [3, 13, 17, 16]. Denote by Cℓm either the real Clifford algebra R0,m or the complex one Cm , generated by the vectors e1 , . . . , em such that e2j = −1 for j = 1, . . . , m. As usual, we identify a vector x = (x1 , . . . , xm ) ∈ Rm with the element x1 e1 + · · · + xm em of the Clifford algebra Cℓm . Let G ⊂ Rm be open. Then a continuously differentiable function f : G → Cℓm is called monogenic if it satisfies the equation ∂f = 0 on G where the Dirac operator ∂ is defined as ∂ = e1 ∂x1 + · · · + em ∂xm .

(7)

Denote by L2 (Bm , Cℓm ) ∩ Ker ∂ the space of L2 -integrable monogenic functions g : Bm → Cℓm . It is well-known that L2 (Bm , Cℓm ) ∩ Ker ∂ forms the right Cℓm -linear Hilbert space. Let us recall a construction of an orthogonal basis in this space, see [20] for more details. It is easy to see that the polynomials monk2 (x1 , x2 ) = (x1 − e12 x2 )k2 /(k2 !), k2 ∈ N0

(8)

form an orthogonal basis of the space L2 (B2 , Cℓ2 ) ∩ Ker ∂. Here we write e12 = e1 e2 as usual. To construct the bases in higher dimensions, we need (k ) (k ) the embedding factors Xm,jm = Xm,jm (x) defined as (k )

Xm,jm =

m − 2 + km + 2j (km ) (km −1) (x) xem , x ∈ Rm Fm,j (x) + Fm,j+1 m − 2 + 2j

(9)

(−1)

(k )

where x = x1 e1 + · · · + xm−1 em−1 , Fm,jm are given in (3) and Fm,j+1 = 0. Then it is well-known that an orthogonal basis of the space L2 (Bm , Cℓm ) ∩ Ker ∂ is formed by the polynomials (k )

m monk (x) = Xm,k ∗

m−1

(k

)

m−1 Xm−1,k ∗

m−2

(k )

· · · X3,k3∗ monk2 (x1 , x2 ) 2

(10)

where k = (k2 , · · · , km ) ∈ Nm−1 and kr∗ = k2 + · · · + kr . Let us remark that 0 due to non-commutativity of the Clifford multiplication the order of factors in the product (10) is important. See [20] for more details. In comparison (k ) with [20], we use another normalization of the embedding factors Xm,jm and we also change the notation for indices to get a nice expression for generating functions.

6

P. Cerejeiras, U. K¨ ahler and R. L´ aviˇcka

Definition 2. We define the generating function Mm of the orthogonal basis monk , k ∈ Nm−1 of spherical monogenics in Rm by 0 X Mm (x, h) = monk (x) hk k∈Nm−1 0

whenever the series on the right-hand side converges absolutely. Here x ∈ Rm and h = (h2 , . . . , hm ) ∈ Rm−1 . In particular, it is easily seen that M2 (x1 , x2 , h2 ) =

∞ X (x1 − e12 x2 )k2 k2 h2 = exp((x1 − e12 x2 )h2 ). k2 !

k2 =0

Here exp((x1 − e12 x2 )h2 ) = exp(x1 h2 )(cos(x2 h2 ) − e12 sin(x2 h2 )). To study the generating functions in higher dimensions we need to know the generating (k ) function of the embedding factors Xm,jm . Lemma 2. We have that ∞ X (k ) Xm,jm (x)hkmm = km =0

1 + xhm em (1 − 2xm hm + h2m |x|2m )m/2+j

where |x|m ≤ 1, |hm | < 1 and j ∈ N0 . Proof. Put ν = m/2 − 1 + j. By (9), the series we want to sum up is equal to ∞ ∞ X X km + 2ν (km ) (km −1) (x)hkmm xem = Σ1 + Σ2 . Fm,j+1 Fm,j (x)hkmm + 2ν

km =0

km =1

Obviously, by Lemma 1, we get that Σ2 =

xhm em . (1 − 2xm hm + h2m |x|2m )ν+1

Moreover, using Lemma 1 again, we have that Σ1 =

∞ 1 hm X (km ) Fm,j (x)km hkmm −1 + 2ν (1 − 2xm hm + h2m |x|2m )ν km =1

and hence Σ1 =

hm d 1 1 + , 2ν dhm (1 − 2xm hm + h2m |x|2m )ν (1 − 2xm hm + h2m |x|2m )ν

which gives Σ1 =

1 − xm hm . (1 − 2xm hm + h2m |x|2m )ν+1

Finally, we conclude that Σ1 + Σ2 = which finishes the proof.

1 + xhm em , (1 − 2xm hm + h2m |x|2m )m/2+j 

Generating functions

7

Now we can prove basic properties of the generating functions Mm quite similarly as in the harmonic case if, in this case, we use Lemma 2 instead of Lemma 1. Then we obtain the following result. Theorem 2. For each m ≥ 2 there is a neighborhood Um of 0 in Rm−1 such that the following statements hold true. (i) The generating functions Mm (x, h) are defined if |x|m ≤ 1 and h ∈ Um . (ii) For each k ∈ Nm−1 , we have that 0 1 k ∂ Mm (x, h)|h=0 , |x|m ≤ 1 k!

monk (x) =

m where k! = (k2 !) · · · (km !) and ∂ k = ∂hk22 · · · ∂hkm . (iii) For m ≥ 3, |x|m ≤ 1 and h ∈ Um , we have that

−m

Mm (x, h) = (1 + xhm em ) dm 2 Mm−1 (x, h/dm ) where dm = 1 − 2xm hm + h2m |x|2m , x = (x1 , · · · , xm−1 ) and h/dm = (h2 /dm , · · · , hm−1 /dm ). Using the recurrence formula (iii) of Theorem 2, we can find closed formulæ of generating functions for spherical monogenics in Rm by induction on the dimension m. Corollary 2. In particular, we have the following formula   (x1 − e12 x2 )h2 1 + xh3 e3 . exp M3 (x1 , x2 , x3 , h2 , h3 ) = 1 − 2x3 h3 + h23 |x|23 (1 − 2x3 h3 + h23 |x|23 )3/2 Remark 3. If one replaces in the definition of the orthogonal basis (10) the polynomials monk2 (x1 , x2 ) = (x1 − e12 x2 )k2 /(k2 !) with monk2 (x1 , x2 ) = (x1 − e12 x2 )k2 ,

(11)

the corresponding generating functions M m are different from Mm but they obviously satisfy again Theorem 2. In particular, we have that M 2 (x1 , x2 , h2 ) =

∞ X

(x1 − e12 x2 )k2 hk22 =

k2 =0

1 − (x1 + e12 x2 )h2 . 1 − 2x1 h2 + h22 |x|22

Acknowledgements We would like to thank V. Souˇcek for useful discussions on this topic. The work of the first and second 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 - Funda¸ca˜o para a Ciˆencia e a Tecnologia”), within project PEst-OE/MAT/UI4106/2014.

8

P. Cerejeiras, U. K¨ ahler and R. L´ aviˇcka

References [1] S. Bock and K. G¨ urlebeck, On a generalized Appell system and monogenic power series, Math. Methods Appl. Sci. 33 (2010) 394–411. [2] S. Bock, K. G¨ urlebeck, R. L´ aviˇcka, V. Souˇcek, The Gel’fand-Tsetlin bases for spherical monogenics in dimension 3, Rev. Mat. Iberoamericana 28 (2012) (4), 1165-1192. [3] F. Brackx, R. Delanghe, F. Sommen, Clifford analysis, Pitman, London, 1982. [4] F. Brackx, H. De Schepper, R. L´ aviˇcka, V. Souˇcek, Gelfand-Tsetlin Bases of Orthogonal Polynomials in Hermitean Clifford Analysis, Math. Methods Appl. Sci. 34 (2011), 2167-2180. [5] F. Brackx, H. De Schepper, R. L´ aviˇcka, V. Souˇcek, Embedding Factors for Branching in Hermitian Clifford Analysis, to appear in Complex Anal. Oper. Theory [6] I. Ca¸c˜ ao, Constructive approximation by monogenic polynomials, Ph.D thesis, Univ. Aveiro, 2004. [7] I. Ca¸c˜ ao, K. G¨ urlebeck, S. Bock, On derivatives of spherical monogenics, Complex Var. Elliptic Equ. 51 (2006)(811), 847–869. [8] I. Ca¸c˜ ao, K. G¨ urlebeck, S. Bock, Complete orthonormal systems of spherical monogenics - a constructive approach, in: L.H. Son, W. Tutschke, S. Jain (Eds.), Methods of Complex and Clifford Analysis, Proceedings of ICAM, Hanoi, SAS International Publications, 2004. [9] I. Ca¸c˜ ao, K. G¨ urlebeck, H.R. Malonek, Special monogenic polynomials and L2 approximation, Adv. appl. Clifford alg. 11 (2001)(S2) 47–60. [10] I. Ca¸c˜ ao and H. R. Malonek, Remarks on some properties of monogenic polynomials, ICNAAM 2006. International conference on numerical analysis and applied mathematics 2006 (T.E. Simos, G. Psihoyios, and Ch. Tsitouras, eds.), Wiley-VCH, Weinheim, 2006, pp. 596-599. [11] I. Ca¸c˜ ao and H. R. Malonek, On a complete set of hypercomplex Appell polynomials, Proc. ICNAAM 2008, (T. E. Timos, G. Psihoyios, Ch. Tsitouras, Eds.), AIP Conference Proceedings 1048, 647-650. [12] R. Delanghe, R. L´ aviˇcka, V. Souˇcek, The Gelfand-Tsetlin bases for Hodgede Rham systems in Euclidean spaces, Math. Meth. Appl. Sci. 35 (2012) (7), 745-757. [13] R. Delanghe, F. Sommen, V. Souˇcek, Clifford algebra and spinor-valued functions, Kluwer Academic Publishers, Dordrecht, 1992. [14] C.F. Dunkl, Y. Xu, Orthogonal Polynomials of Several Variables, Cambridge University Press, Cambridge, 2001. [15] J. E. Gilbert, M. A. M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, Cambridge University Press, Cambridge, 1991. [16] K. G¨ urlebeck, K. Habetha, W. Spr¨ oßig, Holomorphic functions in the plane and n-dimensional space. Translated from the 2006 German original, with cd-rom (Windows and UNIX), Birkh¨ auser Verlag (Basel, 2008). [17] K. G¨ urlebeck, W. Spr¨ oßig, Quaternionic and Clifford Calculus for Physicists and Engineers, J. Wiley & Sons, Chichester, 1997. [18] S.G. Krantz, H.R. Parks, A Primer of Real Analytic Functions, Birkh¨ auser, Basel, 1992.

Generating functions

9

[19] R. L´ aviˇcka, Canonical bases for sl(2,C)-modules of spherical monogenics in dimension 3, Arch. Math.(Brno) 46 (2010) (5), 339-349. [20] R. L´ aviˇcka, Complete orthogonal Appell systems for spherical monogenics, Complex Anal. Oper. Theory 6 (2012) (2), 477489. [21] R. L´ aviˇcka, Orthogonal Appell bases for Hodge-de Rham systems in Euclidean spaces, Adv. appl. Clifford alg. 23 (2013) (1), 113-124. [22] R. L´ aviˇcka, Hypercomplex Analysis - Selected Topics, habilitation thesis, Faculty of Mathematics and Physics, Charles University, Prague, 2011. [23] A. I. Molev, Gelfand-Tsetlin bases for classical Lie algebras, in: M. Hazewinkel (Ed.), Handbook of Algebra, Vol. 4, Elsevier, 2006, pp. 109-170. [24] S. Roman, The Umbral Calculus, Academic Press Inc., 1984. [25] F. Sommen, Spingroups and spherical means III, Rend. Circ. Mat. Palermo (2) Suppl. No 1 (1989) 295-323. P. Cerejeiras CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus de Santiago, P 3810-193 Aveiro, Portugal e-mail: [email protected] U. K¨ ahler CIDMA - Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, Campus de Santiago, P 3810-193 Aveiro, Portugal e-mail: [email protected] R. L´ aviˇcka Faculty of Mathematics and Physics, Charles University in Prague, Sokolovsk´ a 83, 186 75 Praha 8, Czech Republic e-mail: [email protected]