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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: yua@rambler.ru

© 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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36 Yu. A. Brychkov

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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38 Yu. A. Brychkov

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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Integral Transforms and Special Functions 41

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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