On some properties of the Nuttall function Q μ, ν ( ...
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On some properties of the Nuttallfunction Qµ, ν(a, b)Yu. A. Brychkova
a Dorodnicyn Computing Center of the Russian Academy ofSciences, Vavilov str. 40, Moscow 119333, V-333, RussiaPublished online: 29 Jul 2013.
To cite this article: Yu. A. Brychkov (2014) On some properties of the Nuttall function Qµ, ν(a, b),Integral Transforms and Special Functions, 25:1, 34-43, DOI: 10.1080/10652469.2013.812172
To link to this article: http://dx.doi.org/10.1080/10652469.2013.812172
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Integral Transforms and Special Functions, 2014Vol. 25, No. 1, 34–43, http://dx.doi.org/10.1080/10652469.2013.812172
On some properties of the Nuttall function Qμ,ν(a, b)
Yu. A. Brychkov*
Dorodnicyn Computing Center of the Russian Academy of Sciences, Vavilov str. 40, Moscow 119333,V-333, Russia
(In final form 2 July 2013)
Differentiation formulas for the Nuttall function Qμ,ν(a, b) with respect to a and b, generating functions,a closed expression with integer ν in terms of a confluent Appell function, and other relations are given.
Keywords: special functions; Nuttall function; Marcum Q function; differentiation formulas; generatingfunctions; integrals; series
Mathematics Subject Classification: 33E20
The Nuttall function Qμ,ν(a, b) is defined by the integral
Qμ,ν(a, b) =∫ ∞
bxμ e−(x2+a2)/2Iν(ax) dx, (1)
where Iν(z) is the modified Bessel function, b > 0. Applications include the error proba-bility performance of noncoherent digital communication, the outage probability of wirelesscommunication systems, the performance analysis and capacity statistics of uncoded multiple-input multiple-output systems and so on (see [1,2]). A special case is the Marcum functionQν(a, b) = a1−νQν,ν−1(a, b) which is also of interest for applications; it was studied in a numberof papers (see, for example [2–4], and the references there). The properties of Qμ,ν(a, b) werepartly studied in [1,2,5–7]. In particular, a recursion-type relation [6]
Qμ,ν(a, b) = aQμ−1,ν+1(a, b) + (μ + ν − 1)Qμ−2,ν(a, b) + bμ−1 e−(a2+b2)/2Iν(ab),
a sum representation in terms of the Marcum function [7] (if μ and ν are positive integers andμ + ν is odd) and in terms of the incomplete gamma function [1] (if μ and ν are half-integer)were obtained.
In this note we derive some new relations for the Nuttall function, including differentialformulas, integrals and series.
*Email: [email protected]
© 2013 Taylor & Francis
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1. Recurrence-type relation. A recurrence-type relation for Qμ,ν(a, b)
Qμ,ν(a, b) = 2(ν + 1)
aQμ−1,ν+1(a, b) + Qμ,ν+2(a, b) (2)
follows from the known recurrence relation [8, 7.11.(23)] for the Bessel functions
Iν(z) = 2(ν + 1)
zIν+1(z) + Iν+2(z).
2. Distant recurrence-type relations with respect to ν. Using the distant recurrence relationsfor Iν(z) [9, 7.2.4.14–15], we obtain the formulas
Qμ,ν−n(a, b) = (−2)n−1a−n(1 − ν)n−1
[a
�(n−1)/2�∑k=0
(n − k
k
)(−a2/4)k
(1 − ν)k(ν − n + 1)k
Q2k+μ−n+1,ν+1(a, b) + 2ν
�n/2�∑k=0
(n − k
k
)(−a2/4)k
(−ν)k(ν − n + 1)kQ2k+μ−n,ν(a, b)
]
(3)
and
Qμ,ν+n(a, b) = (−2)n−1a−n(ν + 1)n−1
[a
�(n−1)/2�∑k=0
(n − k − 1
k
)(−a2/4)k
(ν + 1)k(1 − n − ν)k
Q2k+μ−n+1,ν−1(a, b) − 2ν
�n/2�∑k=0
(n − k
k
)(−a2/4)k
(ν)k(1 − n − ν)kQ2k+μ−n,ν(a, b)
](4)
for n ≥ 1.3. Formulas of differentiation. The relation (1) can be written as
a−(μ+1)/2 e1/(2a)Qμ,ν
(1√a
,√
ab
)=
∫ ∞
bxμ e−ax2/2Iν(x) dx.
Differentiating with respect to a, we obtain
Dna
[a−(μ+1)/2 e1/(2a)Qμ,ν
(1√a
,√
ab
)]=
(−1
2
)n
a−(μ+1)/2−n e1/(2a)Qμ+2n,ν
(1√a
,√
ab
).
(5)It is known that if
Dnz [f (z)] = F(z),
then
Dnz
[zn−1f
(1
z
)]= (−1)nz−n−1F
(1
z
).
Applying this property to (5) we obtain
Dna
[an+(μ−1)/2 ea/2Qμ,ν
(√a,
b√a
)]= 1
2na(μ−1)/2 ea/2Qμ+2n,ν
(√a,
b√a
). (6)
The formula of differentiation [9, 1.13.1.1]
Dnz [Iν(z)] = 2−n
n∑k=0
(nk
)Iν−n+2k(z)
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results in
Dna[ea2/2Qμ,ν(a, b)] = 2−n ea2/2
n∑k=0
(nk
)Qμ+n,ν−n+2k(a, b). (7)
From the equation
Dna[Qμ,ν(a, b)] = Dn
z [e−a2/2 ea2/2Qμ,ν(a, b)] =n∑
k=0
(nk
)Dn−k
a [e−a2/2]Dkz [ea2/2Qμ,ν(a, b)],
after some calculations, we obtain the formula
Dna[Qμ,ν(a, b)] = (−1)n2−n/2
n∑k=0
(−1)k
(nk
)2−k/2Hn−k
(a√2
)
×k∑
p=0
(kp
)Qμ+k,ν−k+2p(a, b), (8)
where Hn(√
cz) is the Hermite polynomial, and the relation
Dnz [e−cz2 ] = (−1)ncn/2 e−cz2
Hn(√
cz)
was used. The first derivative has the form
Da[Qμ,ν(a, b)] = −aQμ,ν(a, b) + 12 [Qμ+1,ν−1(a, b) + Qμ+1,ν+1(a, b)].
The next differentiation formula can be derived as follows. From (1) we have
ea/2Qμ,ν(√
a, b) =∫ ∞
bxμ e−x2/2Iν(
√ax) dx. (9)
Using the relation [8, 7.11(19)]
Dnz [zν/2Iν(c
√z)] =
( c
2
)nz(ν−n)/2Iν−n(c
√z)($%$) (10)
we get
Dna[aν/2 ea/2Qμ,ν(
√a, b)] = 1
2na(ν−n)/2 ea/2Qμ+n,ν−n(
√a, b). (11)
Now
Dna[Qμ,ν(
√a, b)] = Dn
a[a−ν/2 e−a/2aν/2 ea/2Qμ,ν(√
a, b)]
=n∑
k=0
(nk
)Dn−k
a [a−ν/2 e−a/2]Dka[aν/2ea/2Qμ,ν(
√a, b)],
whence we obtain
Dna[Qμ,ν(
√a, b)] = n!a−n
n∑k=0
2−kak/2
k! L−ν/2−n+kn−k
(a
2
)Qμ+k,ν−k(
√a, b), (12)
where Lλn (z) is the Laguerre polynomial and the relation [9, 10.12(5)]
Dnz [zλ e−az] = n!zλ−n e−azLλ−n
n (z)
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was used. Now we can apply the formula
Dnz [f (z2)] = n!
[n/2]∑k=0
(2z)n−2k
k!(n − 2k)! f (n−k)(z2), (13)
f (n)(z2) = Dnw[f (w)]|w=z2 , to obtain the relation
Dna[Qμ,ν(a, b)] = n!
(2
a
)n [n/2]∑k=0
(n − k
k
)2−2k
×n−k∑p=0
(a/2)p
p! L−ν/2−n+k+pn−k−p
(a2
2
)Qμ+p,ν−p(a, b), (14)
The first derivative has the form
Da[Qμ,ν(a, b)] = Qμ+1,ν+1(a, b) + ν − a2
aQμ,ν(a, b).
Application to (9) of the relation [8, 7.11(20)]
Dnz [z−ν/2Iν(c
√z)] =
(− c
2
)nz(−ν−n)/2Iν+n(c
√z) (15)
results in
Dna[a−ν/2 ea/2Qμ,ν(
√a, b)] = 1
2na−(ν+n)/2 ea/2Qμ+n,ν+n(
√a, b) (16)
and
Dna[Qμ,ν(a, b)] = n!
(2
a
)n [n/2]∑k=0
(n − k
k
)2−2k
×n−k∑p=0
2−pap
p! Lν/2−n+k+pn−k−p
(a2
2
)Qμ+p,ν+p(a, b). (17)
Other differentiation formulas are
Dna[ea/2Qμ,ν(
√a, b)] = (−a)−n ea/2
n∑k=0
(nk
) (−ν
2
)n−k
(−
√a
2
)k
Qμ+k,ν−k(√
a, b), (18)
Dna[Qμ,ν(
√a, b)] =
(−1
2
)n n∑k=0
(nk
) (2
a
)k k∑p=0
(kp
) (ν
2
)k−p
×(
−√
a
2
)p
Qμ+p,ν−p(√
a, b), (19)
whence, using (13), one can obtain the derivatives Dna[Qμ,ν(a, b)].
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Now we evaluate derivatives with respect to b. From the definition (1) we have
Dn+1b [Qμ,ν(a,
√b)] = −e−a2/2
2Dn
b[b(μ−1)/2 e−b/2Iν(a√
b)]
= −e−a2/2
2
n∑k=0
(nk
)Dn−k
b [b(μ−ν−1)/2 e−b/2]Dnb[bν/2Iν(a
√b)],
whence, due to the formula (10),
Dn+1b [Qμ,ν(a,
√b)] = −n!
2b(μ−1)/2−n e−(a2+b)/2
×n∑
k=0
(a√
b/2)k
k! L(μ+ν−1)/2−n+kn−k
(b
2
)Ik+ν(a
√b) (20)
and, by virtue of (13),
Dnb[Qμ,ν(a, b)] = −21−nn!aμ−n+1 e−(a2+b2)/2
n−1∑k=0
22kk!(n − k − 1)!(2k − n + 2)!
×k∑
p=0
(ab/2)p
p! L(μ−ν−1)/2−k+pk−p
(b2
2
)Iν−p(ab), n ≥ 1. (21)
The differentiation formula (15) gives also another expression:
Dnb[Qμ,ν(a, b)] = −21−nn!aμ−n+1 e−(a2+b2)/2
n−1∑k=0
22kk!(n − k − 1)!(2k − n + 2)!
×k∑
p=0
(ab/2)p
p! L(μ+ν−1)/2−k+pk−p
(b2
2
)Iν+p(ab), n ≥ 1. (22)
4. Representations. Starting from (1) and using the expansion
In+1/2(z) = 1√2πaz
[ez
n∑k=0
(−1)k(n + k)!k!(n − k)!zk
+ (−1)n+1 e−zn∑
k=0
(n + k)!k!(n − k)!zk
],
we obtain
Qm+1/2,n+1/2(a, b) = 1
2
n∑k=0
(n + k)!k!(n − k)!(2a)k
{[(−1)k + (−1)n]Qm−k+1/2,1/2(a, b)
+ [(−1)k − (−1)n]Qm−k+1/2,−1/2(a, b)}, (23)
where m, n are positive integers. The functions in the right-hand side can be reduced to derivativesof the integrals ∫ ∞
beax−x2/2 dx =
√π
2ea2/2 erfc
(b − a√
2
)
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and evaluated in terms of the complimentary error function erfc(z) and Hermite polynomialsHn(z). After some calculations, we obtain for m ≥ n the representation
Qm+1/2,n+1/2(a, b) = 2−m/2−1
√a
i−Mn∑
k=0
(k + n)!k!(n − k)! (i
√2a)−k
×{[
erfc
(b − a√
2
)− (−1)n erfc
(a + b√
2
)]Hm−k
(ia√
2
)
− 2√π
m−k∑p=1
(m − k
p
)i−pHm−k−p
(ia√
2
)
×[(−1)n e−(a+b)2/2Hp−1
(a + b√
2
)+ e−(a−b)2/2Hp−1
(a − b√
2
)]}, (24)
which is simpler than that given in [6].5. Power expansion. Expanding the Bessel function Iν(ax) in (1) into power series with respect
to ax we get
Qμ,ν(a, b) =∫ ∞
bxμ e−(x2+a2)/2Iν(ax) dx
= e−a2/2∫ ∞
bxμ e−x2/2
(ax
2
)ν∞∑
k=0
(ax)2k
4kk!�(k + ν + 1)dx
= e−a2/2∫ ∞
bxμ
(ax
2
)ν
e−x2/2∞∑
k=0
(ax)2k
4kk!�(k + ν + 1)dx
=(a
2
)ν
e−a2/2∞∑
k=0
(a)2k
4kk!�(k + ν + 1)
∫ ∞
bx2k+μ+ν e−x2/2 dx
= 2(m−ν−1)/2aν
�(ν + 1)e−a2/2
∞∑k=0
(a2/2)k
k!(ν + 1)k�
(k + m + ν + 1
2,
b2
2
).
We can now expand e−a2/2 into the power series, and obtain, after multiplication, the followingseries expansion:
Qμ,ν(a, b) =∞∑
k=0
Ak(μ, ν, b)a2k+ν , (25)
where
Ak(μ, ν, b) = 2(μ−ν−1)/2
�(ν + 1)
(−1/2)k
k!k∑
p=0
(kp
)(−1)p
(ν + 1)p�
(p + μ + ν + 1
2,
b2
2
).
6. Generating function. If μ ≥ 0 and ν are integers a generating function can be constructedstarting from that for the Bessel functions
∞∑k=−∞
tkIk(z) = e(t+1/t)z/2, t �= 0.
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Substituting z = ax, multiplying by xμ e−x2/2 and integrating from b to ∞ we obtain, in accordancewith the definition (1),
∞∑k=−∞
tkQμ,k(a, b) = e−a2/2∫ ∞
bxμ e−x2/2 e(t+1/t)ax/2 dx, t �= 0. (26)
For μ = 0, the integral has the form
∫ ∞
be−x2/2 e(t+1/t)ax/2 dx =
√π
2exp
[a2(t2 + 1)2
8t2
]erfc
(2bt − a(t2 + 1)
2√
2t
)
For positive integer μ = m, the integral can be evaluated by differentiation with respect to a.Substituting it into (26) we obtain the generating function
∞∑k=−∞
tkQm,k(a, b) = 2(t1−m)/2(−i)m exp
[1
2
(−a2 + ab
(t + 1
t
)− b2
)]
×m∑
k=1
ik
(mk
)Hk−1
(2bt − a(t2 + 1)
2√
2t
)Hm−k
(ia(t2 + 1)
2√
2t
)
+√
π(−i)m
2(m+1)/2exp
[a2(t2 − 1)2
8t2
]erfc
(2bt − a(t2 + 1)
2√
2t
)
× Hm
(ia(t2 + 1)
2√
2t
), t �= 0. (27)
7. Expansion of Qμ,ν(a, b) in series of the Bessel functions. Substituting the expansion[10, 13.13.2]
Iν(ax) = xν
∞∑k=0
(x2 + 1)k
k!(a
2
)kJk+ν(a).
into the definition (1) we can write
Qμ,ν(a, b) =∞∑
k=0
(a/2)k
k! Jk+ν(a)
∫ ∞
bxμ+ν(x2 + 1)k e−(x2+a2)/2 dx
= 1
2
∞∑k=0
(a/2)k
k! e−a2/2Jk+ν(a)
∫ ∞
b2y(μ+ν−1)/2(y + 1)k e−y/2 dy
= 1
2
∞∑k=0
(a/2)k
k! e−a2/2Jk+ν(a)
k∑p=0
(kp
) ∫ ∞
b2yp+(μ+ν−1)/2 e−y/2 dy.
Taking into account the equality
∫ ∞
b2yk+(μ+ν−1)/2 e−y/2 dy = 2k+(μ+ν+1)/2�
(k + μ + ν + 1
2,
b2
2
)
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we get the expansion
Qμ,ν(a, b) = 2(μ+ν−1)/2 e−a2/2∞∑
k=0
(a/2)k
k!
×⎡⎣ k∑
p=0
(kp
)2p�
(p + μ + ν + 1
2,
b2
2
)⎤⎦ Jk+ν(a). (28)
8. Expansion of Qμ,ν(a, b)in series of squares of the Bessel functions. Similarly, using theformula [10, 14.13.2] we can derive the expansion
Qμ,ν(a, b) = 2(μ+3ν−1)/2a−ν e−a2/2�(ν)
∞∑k=0
k + ν
k!
×⎡⎣ k∑
p=0
(kp
)2−p (2ν)k+p
(ν + 1/2)p�
(p + μ + ν + 1
2,
b2
2
)⎤⎦ J2
k+ν(a). (29)
9. Sums and integrals. Now we change μ to 2k + μ in (1), multiply by (−1)k/(2k)! and sumon k from 0 to ∞. We obtain
∞∑k=0
tk
k!Q2k+μ,ν(a, b) =∫ ∞
bxμ e−(x2+a2)/2 e−(1−2t)x2/2Iν(ax) dx
= (1 − 2t)−(μ+1)/2∫ ∞
b√
1−2txμ e−(x2+a2)/2Iν
(ax√
1 − 2t
)dx,
whence we have
∞∑k=0
tk
k!Q2k+μ,ν(a, b) = (1 − 2t)−(μ+1)/2 ea2t/(1−2t)Qμ,ν
(a√
1 − 2t, b
√1 − 2t
),
where |t| < 12 .
A number of series can be obtained by direct application of (1) to known series containingBessel functions Iν(z) [9,11]. For example, from [11, 5.8.3.4]
∞∑k=0
tk
k! Ik+ν(z) =(
2t
z+ 1
)−ν/2
Iν(√
z2 + 2tz)
, |2t| < |z|, | arg z| < π . (30)
we have
∞∑k=0
tk
k!akxkxμ e−(x2+a2)/2Ik+ν(ax) = xμ e−(x2+a2)/2(2t + 1)−ν/2Iν(ax√
2t + 1).
By virtue of the formula (1) we immediately obtain the relation
∞∑k=0
tk
k!Qk+μ,k+ν(a, b) =(
2t
a+ 1
)−ν/2
eatQμ,ν(√
a2 + 2at, b). (31)
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Some other summations and indefinite integrals for Qμ,ν(a, b) are:
∞∑k=0
(k + ν)C(ν)
k (z)Qμ,k+ν(a, b)
= 2−ν−1/2√πaν+1z
�(ν)√
azea2(z2−1)/2[Qμ+ν+1/2,1/2(az, b) + Qμ+ν+1/2,−1/2(az, b)], (32)
∞∑k=0
Tk(z)Qμ,k(a, b)
= −1
2Qμ,0(a, b) +
√πaz
8ea2(z2−1)/2[Qμ+1/2,1/2(az, b) + Qμ+1/2,−1/2(az, b)], (33)
n∑k=0
(nk
)(a/2)k
(ν − n + 1)kQk+μ,k+ν(a, b) = (−a/2)n
(−ν)nQμ+n,ν−n(a, b), (34)
[n/2]∑k=0
(n2k
) (1
2
)k
(2
a
)k
Qμ−k,n−k−1/2(a, b)
= 1
2{[1 + (−1)n]Qμ,−1/2(a, b) + [1 − (−1)n]Qμ,1/2(a, b)}, (35)
n∑k=1
(2k + ν)Qμ,2k+ν(a, b) = a
2[Qμ+1,ν+1(a, b) − Qμ+1,2n+ν+1(a, b)], (36)
n∑k=0
(−1)k
(nk
)(2k + ν)
(ν)k
(n + ν + 1)kQμ,2k+ν(a, b) = (ν)n+1
(2
a
)n
Qμ−n,n+ν(a, b), (37)
∞∑k=0
(−1)k(2k + ν)(ν)k
k! p+3Fq(−k, k + ν, −n, (ap); (bq); z)Qμ,2k+ν,(a, b) (38)
= 2(μ−ν−1)/2aν
�(ν)e−a2/2
n∑k=0
(−1)k
(nk
) (a2z
2
)k
�
(k + μ + ν + 1
2,
b2
2
) ∏pi=0(ai)k∏qj=0(bj)k
, (39)
∫a1−νea2/2Qμ,ν(a, b) da = a1−ν ea2/2Qμ−1,ν−1(a, b), (40)∫aν+1ea2/2Qμ,ν(a, b) da = aν+1 ea2/2Qμ−1,ν+1(a, b), (41)∫ c
0aν+1(c2 − a2)β−1 ea2/2Qμ,ν(a, b) da = 2β−1cβ+ν�(β) ec2/2Qμ−β,ν+β(c, b), ver (42)
Reβ > 0, Reν > −1,∫ ∞
0aν+1 e−a2pQμ,ν(a, b) da = 1
2
(2p + 1)μ−ν−1
p(μ+ν+1)/2�
(μ + ν + 1
2,
b2p
2p + 1
), ver (43)
Rep > 0, Reν > −1, �(ν, z) is the incomplete gamma function.
Acknowledgements
This work was supported by the Russian Foundation for Fundamental Research under Project No. 13-00334.
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