ACTA UNIVERSITATIS APULENSIS
No 17/2009
INTEGRAL AVERAGES OF MULTIPLE POISSON KERNELS
Serap Bulut Abstract. In [3], Galbraith and Green obtained the mean value of the Poisson kernel P (r, θ) for n > −1. In [4], Haruki and Rassias introduced two generalizations of the Poisson kernel which are Q(θ; a, b) and R(θ; a, b, c, d), and obtained the integral value of Q2 (θ; a, b) by means of R(θ; a, b, c, d). Also, they set as an open problem the result of mean values for n ∈ N. This problem is solved by Kim [5]. Then in [1], the author generalized the result of Kim and obtained the mean values of both Q(θ; a, b) and R(θ; a, b, c, d) for all real u. The purpose of this paper is to give integral averages of multiple Poisson kernels, introduced by Spreafico [6], for all real u. Thus we generalize the main result of Spreafico. 2000 Mathematics Subject Classification: Primary 31A10, 31B10; Secondary 33B15, 33C05. Keywords and phrases: Poisson Kernel, Integral Formula, Hypergeometric Function, Pochhammer symbol, Gamma Function.
1.Introduction It is well known that the Poisson kernel in two dimensions is defined by def
P (r, θ) =
1 − r2 , (1 − reiθ )(1 − re−iθ )
and the integral formula 1 Z 2π P (r, θ)dθ = 1 2π 0 151
S. Bulut - Integral averages of multiple Poisson kernels
holds. Here r is a real parameter satisfying |r| < 1. In [4], Haruki and Rassias introduced two generalizations of the Poisson kernel. The first generalization is defined by def
Q(θ; a, b) =
1 − ab , (1 − aeiθ )(1 − be−iθ )
where a, b are complex parameters satisfying |a| < 1 and |b| < 1. The second generalization is defined by R(θ; a, b, c, d) =
(1 −
aeiθ )(1
L(a, b, c, d) , − be−iθ )(1 − ceiθ )(1 − de−iθ )
where a, b, c, d are complex parameters satisfying |a| < 1, |b| < 1, |c| < 1 and |d| < 1 and def
L(a, b, c, d) =
(1 − ab)(1 − ad)(1 − bc)(1 − cd) . 1 − abcd
Then they proved the integral formulas 1 Z 2π Q(θ; a, b)dθ = 1, 2π 0 1 Z 2π R(θ; a, b, c, d)dθ = 1. 2π 0 If we set c = a and d = b in the last integral, then we obtain 1 Z 2π 1 + ab Q(θ; a, b)2 dθ = . 2π 0 1 − ab Afterwards they set the following definition and open problem. For n = 0, 1, 2, . . ., let 1 Z 2π Q(θ; a, b)n+1 dθ, In = 2π 0 def
where a, b are complex parameters satisfying |a| < 1 and |b| < 1. Open Problem. Compute In for n = 2, 3, 4, . . .. 152
S. Bulut - Integral averages of multiple Poisson kernels
In [5], Kim gave a solution to this open problem. In [1], the author generalized In as follows. Theorem A. [1] For any real number u, the following holds 1 Z 2π Ju := Q (θ; a, b)u dθ = (1 − ab)u 2 F1 (u, u; 1; ab) , 2π 0 where 2 F1 is the usual hypergeometric function. Note that, setting a = b = r generalizes the integral 1 Z 2π n+1 P (r, θ)dθ 2π 0 for all real n > −1, ([3]). In [1], the author also gave integral averages of R (θ; a, b, c, d) as X (u)j (u)k (u)l (u)m j k l m 1 Z 2π Ku := R (θ; a, b, c, d)u dθ = L (a, b, c, d)u a b cd . 2π 0 j!k!l!m! j+l=k+m
2.Multiple poisson kernels In [6], Spreafico obtained the integral formula −1 N X bN 1 Z 2π k Pa,b (θ) dθ = 2π 0 k=1 1 − ak bk
N Y
1 , n=1,n6=k (1 − an bk ) (bk − bn )
where a = (an ) and b = (bn ) are two vectors in the complex open unit N -ball and N Y 1 Pa,b (θ) := . iθ −iθ ) n=1 (1 − an e ) (1 − bn e Let us define −1
M (a, b)
−1 bN k := k=1 1 − ak bk N X
N Y
1 . n=1,n6=k (1 − an bk ) (bk − bn )
153
S. Bulut - Integral averages of multiple Poisson kernels
Then multiple Poisson kernel is of the form M (a, b)
PN (θ; a, b) :=
N Q n=1
(1 − an eiθ ) (1 − bn e−iθ )
and satisfies the integral formula 1 Z 2π PN (θ; a, b)dθ = 1. 2π 0 The purpose of this paper is to give integral averages of multiple Poisson kernels PN (θ; a, b), for all real u. Theorem 1. For all real u, the following holds N X Y (u)ln (u)mn ln mn 1 Z 2π u u Mu := PN (θ; a, b) dθ = M (a, b) an b n , 2π 0 ln !mn ! l=m n=1
where l :=
N X
ln
and
m :=
N X
mn .
n=1
n=1
Proof. Let u be any real number. Define the Pochhammer symbol by (u)k :=
Γ(u + k) Γ(u)
(u 6= −n,
n = 0, 1, 2, . . .) ,
where Γ is the gamma function. If u = −n is a nonpositive integer, define (−n)k := (−n) (−n + 1) · · · (−n + k − 1), so that (−n)k = 0 for k = n + 1, n + 2, . . .. Then ∞ X (u)k k 1 = w (|w| < 1) . u (1 − w) k=0 k! For z = eiθ , one computes that 1 Z 2π PN (θ; a, b)u dθ 2π 0 N M (a, b)u Z 2π Y 1 = dθ u iθ −iθ )u 2π 0 n=1 (1 − an e ) (1 − bn e
Mu =
154
S. Bulut - Integral averages of multiple Poisson kernels
=
M (a, b)u Z 2πi |z|=1
dz N Q
z
n=1 u
=
M (a, b) 2πi
Z |z|=1
(1 − an z)u (1 − bn /z)u
N 1 Y
z
n=1
∞ X (u)ln
ln =0
ln !
alnn z ln
∞ n X (u)mn bm n mn =0
mn ! z mn
dz.
Thus we get Mu = M (a, b)u
N X Y (u)ln (u)mn l=m n=1
where l :=
N X
ln
and
ln !mn ! m :=
n=1
n alnn bm n ,
N X
mn .
n=1
Corollary 2. As a consequence of Theorem 1, we have (i) M1 = J1 = 1, for N = 1 and u = 1. (ii) M2 = J2 = 1+ab , for N = 1 and u = 2. 1−ab (iii) M1 = K1 = 1, for N = 2 and u = 1. (iv) M1 = 1, for N = 3 and u = 1 (see [2]). 2.References [1] S. Bulut, Integral averages of two generalizations of the Poisson kernel by Haruki and Rassias, J. Appl. Math. Stochastic Anal. Volume 2008 (2008), Article ID 760214, 6 pages. [2] S. Bulut, A new generalization of the Poisson kernel, Math. J. Okayama Univ. 48 (2006), 173-180. [3] A. S. Galbraith and J. W. Green, A note on the mean value of the Poisson kernel, Bull. Amer. Math. Soc. 53 (1947), 314-320. [4] H. Haruki and Th. M. Rassias, New generalizations of the Poisson kernel, J. Appl. Math. Stochastic Anal. 10 (1997), no. 2, 191-196. [5] B. Kim, The solution of an open problem given by H. Haruki and T. M. Rassias, J. Appl. Math. Stochastic Anal. 12 (1999), no. 3, 261-263. [6] M. Spreafico, Multiple Poisson kernels, Math. J. Okayama Univ. 51 (2009), 177-178.
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S. Bulut - Integral averages of multiple Poisson kernels
Author: Serap Bulut Civil Aviation College Kocaeli University Arslanbey Campus ˙ 41285 Izmit-Kocaeli Turkey e-mail:
[email protected]
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