Analytic and asymptotic properties of generalized Linnik probability densities

Analytic and Asymptotic Properties of Generalized

Linnik Probability Densities

M. Burak Erdogan

_˘

Department of Mathematics, Bilkent Uni¨ersity, 06533, Bilkent, Ankara, Turkey

and I. V. Ostrovskii

Department of Mathematics, Bilkent Uni¨ersity, 06533, Bilkent, Ankara, Turkey, and Institute of Low Temperature Physics and Engineering, Kharko¨, 310164, Ukraine

Submitted by N. H. Bingham

Received August 23, 1996

1. INTRODUCTION w x

In 1953, Linnik 1 proved that the function < <a

w t s 1r 1 q t_a

Ž .

Ž

.

, a g 0, 2

Ž

.

is a characteristic function of a symmetric probability density p . Since_a

 Ž .4

then, the family of symmetric Linnik densities p :_a a g 0, 2 had several

Ž w x.

probabilistic applications see, e.g., 2]7 . Analytic and asymptotic proper-w x

ties of the densities p were studied in 8 ._a w x

In 1984, Klebanov, Maniya, and Melamed 9 introduced the concept of geometric strict stability and proved that the family of geometrically strictly stable densities coincides with the family of densities with charac-teristic functions

yiu sgn t_{< <}a

w_{a , u}

Ž .

t s 1r 1 q e

Ž

t

.

, < <

a g 0, 2 , u F min par2, p y par2 .

Ž

.

Ž

.

555

0022-247Xr98 $25.00

CopyrightQ 1998 by Academic Press All rights of reproduction in any form reserved.

This family is wider than that of symmetric Linnik densities. Analytic and

w x

asymptotic properties of these densities foru / 0 were studied in 10, 11 .

< < Ž .

For u s min par2, p y par2 , these densities appeared in the papers

w x w x

by Laha 12 and Pillai 13 . w x

In 1992, Pakes 14 showed that, in some characterization problems of mathematical statistics, the probability distributions with characteristic functions 1 wa , u , n

Ž .

t s _{yiu sgn t a} n, < < 1q e t

Ž

.

< <

a g 0, 2 , u F min par2, p y par2 , n ) 0,

Ž

.

Ž

.

Ž .

1 play an important role. Therefore, the problem of the study of analytic and Ž . asymptotic properties of the distributions with characteristic function 1 seems to be of interest.

Set

< <

EPD[



Ž

a , u , n : a g 0, 2 , u F min par2, p y par2 , n ) 0 .

.

Ž

.

Ž

.

4

Ž . Ž .

It turns out Theorem 1 below that, for any a, u, n g EPD, the distribu-tion with characteristic funcdistribu-tion w_{a, u , n} is absolutely continuous. We de-note its density by p_{a, u , n}. Evidently, forn s 1 and u s 0, p_{a , u , n} coincides with p . Hence, p_{a a , u , n} may be viewed as a generalization of the symmetric Linnik densities p . We call p_{a a , u , n} the generalized Linnik density.

The aim of our work is to study analytic and asymptotic properties of the

Ž .

generalized Linnik densities p_{a, u , n} for any a, u, n g EPD and to obtain

w x

corresponding generalizations of results of 8, 11 . However, the main

w x

methods of 8, 11 are not applicable to this aim. The fact is that these methods are based on the idea that in the case n s 1, the representation

Ž .

of p_{a, u , n} by the formula 6 below makes it possible to reduce the problem to the study of some Cauchy type integral. Such a reduction seems to be quite impossible in the general casen ) 0. In this work, we use a different

w x

method. It is prompted by an idea used in 15 to study multidimensional generalizations of Linnik densities. One of the advantages of this method is that the proof of our results is much shorter than the proof of less

w x

general results of 8, 11 .

2. STATEMENT OF RESULTS

We begin with the absolute continuity of the distributions with charac-Ž .

teristic function 1 . The following theorem is an analogue of Theorem

w x w x w x

Ž .

THEOREM 1. For any a, u, n g EPD, the probability distribution with

Ž .

the characteristic functionw_{a, u , n} defined by 1 is absolutely continuous and

1 cqi` _" zy1

p_{a , u , n}

Ž

"x s

.

_H

f_{a , u , n}

Ž .

z x dz,

2p i cyi`

ya - c - min an , 1 , x ) 0,

Ž

.

Ž .

2

where

1 sin

Ž

p zr2 " u zra G zra G n y zra

.

Ž

.

Ž

.

"

f_{a , u , n}

Ž .

z s .

Ž .

3

a G n

Ž

.

sinp z G z

Ž .

Both functions fq_{a, u , n} and fy_{a , u , n} in Theorem 1 are analytic outside of the set

`

` `



yqa

4

qs1j



a n q q

Ž

.

4

qs0j q

 4

qs1.

Ž .

4  < < < < 4

Moreover, in any set z: Re z - H, Im z )« , the following bound holds

" _{< <}

f_{a , u , n}

Ž .

z F exp yC



_"

Ž

a , u Im z ,

.

4

Ž .

5

Ž . q y

where C_" a, u is a positive constant. Since the functions f_{a , u , n} and f_{a , u , n}

 Ž .4 Ž .

are analytic in z: ya - Re z - min an, 1 , the integral in 2 does not Ž .

depend on c, under the restrictions mentioned in 2 .

Ž . 4 q

Denote the subset a, u, n g EPD: u G 0 of EPD by EPD . Note

Ž . Ž . Ž . Ž .

that Eqs. 2 and 3 make it obvious that p_{a, u , n} x s p_{a , yu , n} yx for any

Ža, u, n g EPD. Thus, without loss of generality, we can restrict our study.

Ž . q

of p_{a, u , n} to a, u, n g EPD .

To study the analytic properties and non-symmetry of p_{a, u , n} foru / 0, we shall need the following representation of p_{a, u , n}. The representation

w x

was proved by Linnik 1 in the case u s 0, n s 1 and by the first author w x_{11 in the case u / 0, n s 1. Denote by EPD the subset a, u, n g}U _{Ž .}

q ₄ q _Ž

EPD : u - p y par2 of EPD . In what follows except the proof of

. Ž . U

Theorem 1 in Section 3 , we will consider the case a, u, n g EPD only,

Ž . q U Ž .

and will not consider the case a, u, n g EPD _ EPD s a, u, n g

q ₄

EPD : u s p y par2 . We are going to consider the latter case in a

separate paper.

THEOREM2. The density p_{a, u , n} is representable in the form

` yy x

1 e dy

pa , u , n

Ž

"x s

.

_p Im

_H

_{. iuyip a r2 a} n, x) 0.

Ž .

6

Recall that a function f defined on an interval I; R is called

com-pletely monotonic if it is infinitely differentiable on I and, moreover,

Žy1 f.k Ž k ._{Ž .x} G 0 for any x g I and any k s 0, 1, . . . .

The following theorem related to analytic properties of p_{a, u , n} was

w x w x

proved in the caseu s 0, n s 1 in 8 , in the case u / 0, n s 1 in 11 ; part Ž .ii was also proved in 10 under some additional restrictions. Letw x w xxU

denote the greatest integer strictly less than x.

Ž . Ž .

THEOREM 3. i If an F 2, u F min par2, prn y par2 , then both

Ž . Ž .

of the functions p_{a, u , n} "x are completely monotonic on 0, ` . This assertion

Ž . U

ceases to be true for other_¨alues of a, u, n g EPD . Ne_¨ertheless, for any

Ža, u, n g EPD , both of the functions p. U _{a, u , n}Ž"x are differences of two.

Ž .

completely monotonic functions on 0,` .

Ž .ii Suppose sinŽ Žn u " par2 / 0. Then, for k s an , an q.. w xU w xU 1, . . . , w Ž .xU kqn ar2qurp Ž k . y1 lim p x s q`, 7

Ž

.

_{a , u , n}

Ž .

Ž .

q xª0 w Ž_{n ar2yurp}.xU Ž k . y1 lim p x s q`. 8

Ž

.

_{a , u , n}

Ž .

Ž .

y xª0 Ž .

Ifu s par2, then p_{a, u , n} x s 0 for x - 0.

w xU

If an ) 1, then the function p_{a, u , n} is an y 1 times continuously differen-tiable onR and, moreo¨er,

sin

Ž

pr2 y ura k q 1

. Ž

.

Ž k .

pa , u , n

Ž .

0 s _pa B

Ž

Ž

kq 1 ra , n y k q 1 ra

.

Ž

.

.

w xU for ks 0, 1, . . . , an y 1.

Recall that a probability density is called unimodal with mode a if it is

Ž . Ž .

non-decreasing on y`, a and is non-increasing on a, ` . In the case n s 1, the following theorem was proved independently by Kozubowski w x10 and by the first author 11 . It is worthwhile to mention that the proofw x of our theorem is straightforward and based on a different idea than that

w x w x

used in 10 where a difficult Yamazato unimodality theorem 16 was utilized.

THEOREM4. All generalized Linnik densities are unimodal. Moreo¨er, for

< <u F max 0, prn y par2 the mode is zero and for u ) max 0, prn yŽ . Ž .

The following theorem measures the non-symmetry of p_{a, u , n}. Ž .

THEOREM5. i If an F 1, then

p_{a , u , n}

Ž .

x G p_{a , u , n}

Ž

yx ,

.

x) 0.

Ž .ii As a function of u, 0 F u F min par2, pr 2n y par2 ,Ž Ž . .

Ž . Ž .

p_{a, u , n} x increases and p_{a , u , n} yx decreases for any fixed x ) 0 and a, n

such that an F 1.

The following theorem characterizes the asymptotic behaviour of p_{a, u , n}

w x w x

at`. It was proved in 8 in the caseu s 0, n s 1 and proved in 10, 11 in the caseu / 0, n s 1.

Ž . Ž .

THEOREM6. For any a, u, n g int EPD and for any n s 1, 2, 3, . . . ,

q n

1

Ž

y1 sin

.

Ž

paqr2 " u q G n q q

.

Ž

.

_y1yqa

< < p_{a , u , n}

Ž

"x s

.

_Ý

x 2G

Ž

n

.

_qs1 sin

Ž

p qa G yqa G 1 q q

.

Ž

.

Ž

.

< <y1yna q o x

Ž

.

,

Ž .

9 < < as x ª `. Ž . Ž .

COROLLARY1. For any a, u, n g int EPD ,

p_{a , u , n}

Ž

"x

.

n _{y1ya y1ya} < < < < < < s sin

Ž

par2 " u G 1 q a x

.

Ž

.

q o x

Ž

.

, x ª `. 2p 10

Ž

.

Ž . Ž .

COROLLARY2. For any a, u, n g int EPD ,

p_{a , u , n}

Ž .

x sin

Ž

par2 q u

.

lim s .

p yx sin par2 y u

xªq` _{a , u , n}

Ž

.

Ž

.

Ž . Ž . Ž .

COROLLARY3. For any a, u , n , a, u , n g int EPD ,_{1 1 2 2} pa , u , n1 1

Ž .

x n sin par2 q u1

Ž

1

.

lim s .

p x n sin par2 q u

xªq` _{a , u , n2 2}

Ž .

2

Ž

2

.

Ž . Ž .

Note that Theorem 6 remains valid for a, u, n g ­ EPD as well, but

Ž . Ž . Ž .

for a, u, n g ­ EPD , the sum in the right hand side of 9 vanishes Ž .

therefore 9 does not give satisfactory information related to the

be-Ž < <yK

haviour of p_{a, u , n} at ` except that it decreases faster than x for any .

K .

w x

The following phenomenon was discovered in 8 : the structure of series representations of the symmetric Linnik density p depends on the arith-_a

metical nature of a. This phenomenon is valid for p_{a, u , n},u / 0, n s 1, as w x

shown in 11 . We will show that this phenomenon remains in force in the Ž .

general case of p_{a, u , n} x taking into account the arithmetical nature of

two parameters a, n.

Theorems 7, 8, 10, 11 below are generalizations of the corresponding

w x w

ones of 8, 11 . Theorem 9 is a generalization of corresponding ones of 11, x

15 .

THEOREM7. Suppose one of the following conditions is satisfied:

Ž .i a f Q and n g Q; Ž .ii a g Q and n f Q;

Žiii. a f Q, n f Q, and n f a s 2 q q 1 ra y p; q g Z , p g N ; Ž . q 4 Živ. a g Q and n g Q where a, n are representable in the form a s mrn, n s krl where m, n and k, l are relati¨ely prime integers and l does not di¨ide m. Then p_{a , u , n}

Ž

"x

.

1 s lim xa G n

Ž

.

sª` q s

_Ž

y1 sin

_.

_Ž

_Ž

n q q par2 " u G n q q

_{. Ž}

_.

_.

_Ž

_.

Žnqq. =

½

a

_Ý

x sin

Ž

pa n q q G a n q q G 1 q q

Ž

.

.

Ž

Ž

.

.

Ž

.

qs0 qq1

y1 sin p qr2 " u qra G qra

Ž

.

Ž

.

Ž

.

_q

q

_Ý

x

5

,

sin

Ž

p n y qra G q G 1 y n q qra

Ž

.

.

Ž .

Ž

.

Ž .

1FqFnqsq1r2 a

x) 0.

Ž

11

.

This limit is uniform with respect to x on e¨ery compact subset of Rq.

The following corollary of Theorem 7 characterizes the asymptotic behaviour of p_{a, u , n} at 0.

COROLLARY4. Under the conditions of Theorem 7,

¡

sin

Ž

n par2 " u

Ž

.

.

_aŽny1.

x

Ž

1q o 1

Ž .

.

foran - 1,

a sin pan G an

Ž

.

Ž

.

~

pa , u , n

Ž

"x s

.

sin

Ž

pr2 " ura G 1ra

.

Ž

.

1q o 1

Ž .

Ž

.

a G n sin p n y 1ra G 1 y n q 1ra

Ž

.

Ž

Ž

.

.

Ž

.

¢

foran ) 1, as xª 0q.

It is natural to ask whether the limits of each of the two sums in the

Ž . w x

right hand side of 11 exist. Similarly to 8, 11, 15 , it is the case for an uncountable dense subset of EPD. To describe this subset we need Liouville numbers. Recall that an irrational number l is called a Liouville number if, for any rs 2, 3, 4, . . . , there exist p, q g Z, q G 2, such that

p 1

0- l y - _r.

q q

We denote the set of all Liouville numbers by L. By the famous Liouville

Ž w x.

theorem see, e.g., 17, p. 7 , all numbers in L are transcendental.

w x

Moreover 17, p. 8 , the set L has the Lebesgue measure zero. THEOREM8. Suppose one of the following conditions is satisfied:

Ž .i a f Q j L and n g Q; Ž .ii a g Q and n f Q j L;

Žiii. a g Q and n g Q where a, n are representable in the form a s mrn, n s krl where m, n and k, l are relati¨ely prime integers and l does not di_¨ide m.

Then p_{a , u , n}

Ž

"x

.

q

`

1

Ž

y1 sin

.

Ž

Ž

n q q par2 " u G n q q

. Ž

.

.

Ž

.

_aŽnqq.y1

s

_Ý

x

G

Ž

n

.

qs0 sin

Ž

pa n q q G a n q q G 1 q q

Ž

.

.

Ž

Ž

.

.

Ž

.

qq1 `

1

Ž

y1

.

sin

Ž

p qr2 " u qra G qra

.

Ž

.

_qy1

q

_Ý

x ,

a G n

Ž

.

qs1 sin

Ž

p n y qra G q G 1 y n q qra

Ž

.

.

Ž .

Ž

.

x) 0,

Ž

12

.

where both of the series in the right hand side con_¨erge absolutely and

uniformly with respect to x on e¨ery bounded subset of Rq.

Nevertheless, the absolute and uniform convergence of both of the

Ž . Ž .

series in 12 is not valid for all a, u, n satisfying conditions of Theorem 7.

Ž .

THEOREM9. Both of the series in the right hand side of 12 di¨erge on an

uncountable dense subset of EPD.

Now, we shall consider the case when none of the conditions in the statement of Theorem 7 is satisfied. It suffices to consider the following

two cases:

Ž .i a f Q, n s q ra y p for some q g Z , p g NU U U q U Ž .ii a s mrn g Q, n s krl g Q, and l divides m. Note that the Diophantine equation

qsa n q p ,

Ž

.

qg Zq, pg N

Ž

13

.

Ž U U. Ž .

has a unique solution q , p in the case i , and it has infinitely many

Ž . Ž .

solutions in the case ii . Denote by D_{a, n}, the set of all solutions q, p in Ž .

the case ii and set

P_{a , n}s p: q, p g D



Ž

.

_{a , n}, pG 0, q G 1

4

Q_{a , n}s q: q, p g D



Ž

.

_{a , n}, pG 0, q G 1 .

4

Ž .

THEOREM10. If the condition i is satisfied, then p_{a , u , n}

Ž

"x

.

1 s lim xa G n

Ž

.

sª` q s _{y1 sin n q q par2 " u G n q q}

Ž

.

Ž

Ž

. Ž

.

.

Ž

.

_aŽnqq. =

½

a

_Ý

x U sin

Ž

pa n q q G a n q q G 1 q q

Ž

.

.

Ž

Ž

.

.

Ž

.

qs0, q/p qq1

y1 sin p qr2 " u qra G qra

Ž

.

Ž

.

Ž

.

_q

q

_Ý

x

5

sin

Ž

p n y qra G q G 1 y n q qra

Ž

.

.

Ž .

Ž

.

U Ž . 1FqFnqsq1r2 a , q/q pUqqU _U U q y1 y1 x p u p u q

Ž

.

_U q _U

_ž

_ž

"

_/

cos q

_ž

"

_/

G

_{ž /}

p G n G 1 q p

Ž

.

Ž

.

2 a 2 a a 1 _X qU _U p u q G

_{ž /}

sin q

_ž

"

_/

/

a a 2 a pUqqU _U U U q y1 y1 x G q ra sin q pr2 " ura

Ž

.

Ž

.

Ž

.

q _{p G n G 1 q p}U

Ž

.

Ž

.

GX

Ž

1q pU

.

= log x y

ž

_U

/

,

Ž

14

.

G 1 q p

Ž

.

U U _{Ž .}

where q , p in the equation are connected by the formula 13 . The limit in

the right hand side of the equation is uniform with respect to x on e¨ery

Ž .

THEOREM11. If the condition ii is satisfied, then

p_{a , u , n}

Ž

"x

.

q

`

1

Ž

y 1 sin

.

Ž

Ž

n qq par2 " u G n qq

. Ž

.

.

Ž

.

_aŽnqq.y1

s

_Ý

x

G

Ž

n

.

qs0, qfPa , n sin

Ž

pa n q q G a n q q G 1 q q

Ž

.

.

Ž

Ž

.

.

Ž

.

qq1 `

1

Ž

y1

.

sin

Ž

p qr2 " u qra G qra

.

Ž

.

_qy1

q

_Ý

x

a G n

Ž

.

qs1, qfQa , n sin

Ž

p n y qra G q G 1 y n q qra

Ž

.

.

Ž .

Ž

.

p_{qq qy1} y1 x p u p u q

Ž

.

q

_Ý

_ž

_ž

"

_/

cos q

_ž

"

_/

G

_{ž /}

p G n G 1 q p

Ž

.

Ž

.

2 a 2 a a qgQa , n 1 _X q p u q G

_{ž /}

sin q

_ž

"

_/

_/

a a 2 a p_{qq qy1}

y1 x G qra sin q pr2 " ura

Ž

.

Ž

.

Ž

.

q

Ý

_{p G n G 1 q p}

Ž

.

Ž

.

qgQ_{a , n} GX

Ž

1q p

.

= log x y

ž

/

,

Ž

15

.

G 1 q p

Ž

.

where q, p in the third and fourth sums of the equation are connected by the

Ž .

formula 13 . Both of the series in the right hand side con¨erge absolutely and uniformly with respect to x on e¨ery bounded subset of Rq.

Ž . Ž .

Note that the log x term in Eqs. 14 , 15 disappears whenu s "par2. It is possible to obtain asymptotic formulas describing the behaviour of

p_{a, u , n} as xª 0q from Theorems 10 and 11. We present the following two examples: Ž . Ž . Ž . EXAMPLE1 Theorem 10 . a s 1r p q 1 , n s p , u s 1r 2p q 2 . sin

Ž

p p " 1 r 2p q 2

Ž

.

Ž

.

.

_y1rŽpq1. p_{a , u , n}

Ž

"x s

.

₂ x sin

Ž

p r p q 1 G pr p q 1

Ž

.

.

Ž

Ž

.

.

1 q O log

_ž

_/

, xª 0q. x Ž . EXAMPLE2 Theorem 11 . a s 2r3, n s 3r2, u s 1. cos 3

Ž

r2

.

1 p_{a , u , n}

Ž

"x s

.

log q O 1 ,

Ž .

xª 0q. p x

3. REPRESENTATION OF p_{a, u , n} BY A CONTOUR INTEGRAL

Ž .

Proof of Theorem 1. Let us define the function p_{a, u , n} by formulas 2 ,

Ž .3 . Since w_{a, u , n} is a characteristic function 14 , it suffices to prove thatw x Ž .

p_{a, u , n} belongs to L1R and its Fourier transform coincides with wa , u , n. Ž .

If we take c positive in 2 , then estimating the integral with the help of Ž .

the bound 5 , we have

< <cy1 _{< <} pa , u , n

Ž .

x s O x

Ž

.

, x ª 0.

If we take c negative, then we have

< <y< c<y1 _{< <}

pa , u , n

Ž .

x s O x

Ž

.

, x ª `.

Ž . Thus p_{a, u , n}g L R .₁

Now, let us prove that the Fourier transform of p_{a, u , n} coincides with w_{a,u , n}. It suffices to prove this for the case t) 0 since the case t - 0 can

Ž . Ž .

be reduced to the first one by using the equalities w_{a, u , n} t sw_{a , yu , n} yt ,

Ž . Ž . p_{a, u , n} x s p_{a , yu , n} yx . We have ` i t x I t

Ž .

[

_H

p_{a , u , n}

Ž .

x e dx y` ` ` 1 _{yi t x} 1 _{i t x} s

ž

_H

q

_H

/

p_{a , u , n}

Ž

yx e

.

dxq

ž

_H

q

_H

/

p_{a , u , n}

Ž .

x e dx. 0 1 0 1 Ž . Ž .

Let 0-« - min a, ab, 1 . Using the equality 2 , we obtain

1 1 _{yi t x} «qi` _{y zy1} I t

Ž .

s

_H

e dx

_H

f_{a , u , n}

Ž .

z x dz 2p i 0 «yi` ` 1 _{yi t x} y«qi` _{y zy1} q

_H

e dx

_H

f_{a , u , n}

Ž .

z x dz 2p i 1 y«yi` 1 1 <sub>i t x</sub> «qi` _{q zy1} q

_H

e dx

_H

f_{a , u , n}

Ž .

z x dz 2p i 0 «yi` ` 1 _y«qi` i t x q zy1 q

_H

e dx

_H

f_{a , u , n}

Ž .

z x dz. 2p i 1 y«yi`

In all integrals in the right hand side, we can change the order of Ž .

integration by using 5 and Fubini’s theorem. Thus

1 «qi` _y 1 _{yit x zy1}

I t

Ž .

s

_H

f_{a , u , n}

Ž .

z dz

_H

e x dx

2p i «yi` 0

`

1 _{y«qi` y yit x zy1}

q

_H

f_{a , u , n}

Ž .

z dz

_H

e x dx 2p i y«yi` 1 1 <sub>«qi` q</sub> 1 _{i t x zy1} q

_H

f_{a , u , n}

Ž .

z dz

_H

e x dx 2p i <sub>«yi`</sub> 0 ` 1 y«qi` _{q i t x zy1} q

_H

f_{a , u , n}

Ž .

z dz

_H

e x dx.

Ž

16

.

2p i y«yi` 1

Both of the integrals H`eyi t xxzy1<sub>dx and H</sub>`_ei t x_xzy1_{dx converge}

uni-1 1

 4

formly on any compact set lying in z: Re z- 1 . Using the equality

` e" it zy 1 `

" it x zy1 " it x zy2 _{< < < <}

e x dxs . . e x dxs O z ,

Ž

.

z ª `,

H

_{it it}

H

1 1

the integrations in the second and fourth integrals in the right hand side of Ž16 can be translated from z: Re z.  s y« to Re z s « . Therefore, the4  4

Ž .

equality 16 can be rewritten in the form

` 1 <sub>«qi`</sub> y yit x zy1 I t

Ž .

s _{2p i}

H

fa , u , n

Ž .

z dz

H

e x dx «yi` 0 ` 1 «qi` _{q i t x zy1} q

_H

f_{a , u , n}

Ž .

z dz

_H

e x dx.

Ž

17

.

2p i «yi` 0

Using the equalities

` _{" it x zy1 yz " ip zr2} e x dxs t G z e

Ž .

, 0- Re z - 1,

H

0 Ž .3 and 17 , we obtainŽ . 1 _«qi` i zu ra yz I t

Ž .

s

_H

G zra G n y zra e

Ž

.

Ž

.

t dz.

Ž

18

.

2p ia G n

Ž

.

«yi`

If 0- t - 1, then, evaluating the integral with the same integrand as in Ž18 along the boundary of the region z: z.  < <- n q 1r2Ž .a, Re z - « by4

Cauchy’s residue theorem, and then letting nª `, we obtain

`

1

i zu ra yz I t

Ž .

s _{a G n}

Ý

Reszsyqa

Ž

G zra G n y zra e

Ž

.

Ž

.

t

.

Ž

.

_qs0 q ` 1

Ž

y1 G

.

Ž

n q q

.

_{yiu q a q} s

_Ý

e t G

Ž

n

.

_qs0 G 1 q q

Ž

.

n yiu a s 1r 1 q e

Ž

t

.

sw_{a , u , n}

Ž .

t . Ž . If t) 1, then evaluating the integral with the same integrand as in 18

 < < Ž . 4

along the boundary of the region z: z - n q n q 1r2 a, Re z ) « , and then letting nª `, we obtain

`

y1 _{i zu ra yz}

I t

Ž .

s _{a G n}

Ý

ReszsŽnqq.a

Ž

G zra G n y zra e

Ž

.

Ž

.

t

.

Ž

.

_qs0 q ` 1

Ž

y1 G

.

Ž

n q q

.

_{iu Žnqq. ya Žnqq.} s

_Ý

e t G

Ž

n

.

qs0 G 1 q q

Ž

.

n yiu a s 1r 1 q e

Ž

t

.

sw_{a , u , n}

Ž .

t . Ž .

Proof of Theorem 2. It suffices to prove that if we define p_{a, u , n} by 6 , Ž .

then its Fourier Transform coincides with w_{a, u , n} t . Set ` _{i t x}

H t

Ž .

[

_H

e p_{a , u , n}

Ž .

x dx. y`

Ž . Ž .

Let us substitute p_{a, u , n} x from 6 . Changing the order of integrals by

Fubini’s Theorem, we derive

` 1 y dy H t

Ž .

s Im

½

_H

_{yiuyip a r2 a} n _{2 2}

5

p 0

_Ž

1q e y

_{. Ž}

y q t

_.

` it dy q Im

½

_H

_n

5

yiuyip a r2 a 2 2 p 0

_Ž

1q e y

_{. Ž}

y q t

_.

` 1 y dy q Im

½

_H

_n

5

iuyip a r2 a 2 2 p 0

_Ž

1q e y

_{. Ž}

y q t

_.

` it dy y Im

½

_H

_n

5

iuyip a r2 a 2 2 p 0

_Ž

1q e y

_{. Ž}

y q t

_.

1

w

x

\ Im Aq it Im B q Im C y it Im D .

Ž

19

.

p

In the complex y-plane, we consider the region

< < < <



4

GRs y sj q ih: y - R, h ) 0 , R) t

and define the branch of the multivalued function ya as

a _{< <}a ia arg y

y s y e , 0F arg y Fp .

The integrands of A and C are analytic in the closure of G except the_R < <

simple pole at ys i t . By Cauchy’s residue theorem we have

y dy p i s 2p i Re s s .

E

_{­ G 1q e}yiuyip a r2 a_y n _y2_{q t}2 i< t < _{1q e}yiu< <_t a n

Ž

.

Ž

. Ž

.

R

Letting Rª `, and using the notation A and C, we obtain 1

Im Aq Im C sp Re _{yiu a} n.

Ž

20

.

< <

1q e t

Ž

.

We have in the similar way

p 1 Im By Im D s _{< <t} Im _{yiu a} n.

Ž

21

.

< < 1q e t

Ž

.

Ž . Ž . Ž .

Substituting 20 and 21 into 19 , we have

1 1 H t

Ž .

s Re _{yiu a} n q i sgn t Im _{yiu a} n < < < < 1q e t 1q e t

Ž

.

Ž

.

1 s

_Ž

_{1q e}yiu sgn t_{< <t} a

_.

n swa , u , n

Ž .

t . Ž . 4. SOME PROPERTIES OF p_{a, u , n} x Ž . Ž .

Proof of Theorem 3. i Equation 6 is equivalent to

` yy x " iuqip a r2 a 1 e sin

Ž

n arg 1 q e

Ž

y

.

.

dy p_{a , u , n}

Ž

"x s

.

_H

_{. iuyip a r2 a n} , x) 0. < < p 0 1q e y 22

Ž

.

Hence ` <sub>yy x 1,"</sub> ` _{yy x 2,"} p_{a , u , n}

Ž

"x s

.

H

e g_{a , u , n}

Ž .

y dyy

H

e g_{a , u , n}

Ž .

y dy, x) 0, 0 0 23

Ž

.

1," _{Ž .} 2," _{Ž .}

where g_{a, u , n} y , g_{a , u , n} y are non-negative, bounded functions. Evidently

Ž .

both of the integrals in the right hand side of Eq. 23 are infinitely differentiable and both are completely monotonic for x) 0.

< < Ž " iuqip a r2 a.

Note that, for u F par2, arg 1 q e y is an increasing func-tion of y) 0. It takes value 0 for y s 0 and takes value "u q par2 for

Ž .

ys `. Therefore, if an F 2, u F min par2, prn y par2 , then the

func-Ž Ž " iuqip a r2 a..

tion sin n arg 1 q e y is non-negative, which completes the proof.

Ž .ii Taking the k th derivative of both sides of Eq. 23 , and thenŽ . applying the monotone convergence theorem, we obtain

Žk. _` k iuqip a r2 a y1 y sin n arg 1 q e y dy

Ž

.

Ž

Ž

.

.

Ž k . lim pq a , u , n

Ž .

x s p

H

<₁q eyiuyip a r2 a_y <n , xª0 0 ` k yiuqip a r2 a 1 y sin

_Ž

n arg 1 q e

_Ž

y

_.

_.

dy Ž k . lim p_{a , u , n}

Ž .

x s

_H

_{iuyip a r2 a n} . y p <₁q e _y < xª0 0

Evidently, the integrals in the right hand side are divergent for ks w_{an , an q 1, . . . , and convergent for k s 0, 1, . . . , an y 1.}xU _{w x}U _{w x}U

In the first case, we have

w Ž .xU kqn ar2qurp Ž k . y1 lim p x

Ž

.

_{a , u , n}

Ž .

q xª0 w Žn ar2qurp.xU _{k iuqip a r2 a} c ` y1 y sin n arg 1 q e y dy

Ž

.

Ž

Ž

.

.

s

½

_H

q

_H

5

_{yiuyip a r2 a n} , < < p 0 c 1q e y Ž Ž iuqip a r2 a..

where c is the last point where sinn arg 1 q e y changes its sign. The first integral in the right hand side is convergent and finite and

Ž . Ž .

the second one is equal to q`, which proves 7 . The proof of 8 is Ž . similar. The validity of the assertion that if u s par2, then p_{a, u , n} x s 0

for x- 0 is obvious.

In the second case, we have

k k ` .1 y dy

Ž

.

Ž k . I"[ lim p_" a , u , n

Ž .

x s _p Im

H

_{. iuyip a r2 a} n 0 xª0

_Ž

1q e y

_.

k _y1qŽkq1.ra ` .1 u du

Ž

.

s Im

_H

_n. . iuyip a r2 pa 0

Ž

1q e u

.

Changing the contour of integration in the last integral from Rq to us re" iuqip a r2: rg R , we haveq4 ` y1qŽkq1.ra sin

Ž

Ž

pr2 y ura k q 1

. Ž

.

.

r dr I_"s

_H

_n pa 0

Ž

1q r

.

sin

Ž

Ž

pr2 y ura k q 1

. Ž

.

.

kq 1 kq 1 s B

_ž

,n y

_/

. pa a a

To prove Theorem 4, we shall use the following theorem.

w x w .

THEOREM 18, p. 48 . Let f be a real _¨alued function defined on 0,` ha_¨ing finitely many, m say, changes of sign. Assume that the integral

` _{yx s}

p x

Ž .

s

_H

e f s ds,

Ž .

0

con¨erges for any x) 0. Then the number of zeros of the function p does not exceed m.

Proof of Theorem 4. Firstly we shall prove the unimodality. Note from

Ž .

representation 6 that the probability density p_{a, u , n} is a continuous

Ž .

function with respect to parameters a, u, n, for fixed x g R and a, u, n g EPDU_{. Consider the following dense subset of EPD}U

Ss



Ž

a , u , n g EPD

.

U: 0-u - par2, urpa f Q .

4

Since any probability density which is a limit of unimodal probability

Ž .

densities is unimodal, it suffices to prove that for any a, u, n g S, the density p_{a, u , n} is unimodal.

Ž . Ž . Ž .

Case i . an ) 2. By Theorem 3 ii , for any a, u, n g S and for any

w xU Ž k . _{Ž .} w x

ks 0, 1, 2, . . . , an y 1, we have p_{a, u , n} 0 / 0. By the Theorem of 18

Ž . w xU

mentioned above and the formula 22 , for any ks 0, 1, . . . , an y 1, the

Ž k . w Ž _. xU w _.

number of zeros of p_{a, u , n} does not exceed n par2 q u rp on 0,`

w Ž . xU Ž x

and does not exceed n par2 y u rp on y`, 0 . Thus the total

Ž k . w xU

number of zeros of any derivative p_{a, u , n}, ks 0, 1, . . . , an y 1, does not

w Ž . xU w Ž . xU

exceed n par2 q u rp q n par2 y u rp . Evidently, this sum w xU

does not exceed an .

Assume, p_{a, u , n} is not unimodal. Then p_{a , u , n}X has at least 3 zeros. By Rolle’s Theorem, p_{a, u , n}Y has at least 4 zeros. Repeated application of

wŽan xU_{y1 .}

w xU

Rolle’s Theorem enables us to say that p_{a, u , n} has at least an y 1 q w xU

2s an q 1 zeros which is impossible.

Ž . X

Case ii . an F 2. By a similar argument, the number of zeros of p_{a, u , n}

w Ž . xU Ž .

w Žn par2 y u rp s 0 on y`, 0 . Evidently p. xU Ž . _{a, u , n}X Ž .x is unbounded as

X _{Ž .} X _{Ž .}

y

xª 0". Since p_{a, u , n} has no zeros on y`, 0 , we have lim_{xª 0} p_{a , u , n} x

X _{Ž .}

q

s q`. We have two cases: limxª 0 pa , u , n x s q` and

X _{Ž .}

q

limxª 0 pa , u , n x s y`, but it is easy to see that pa , u , n is unimodal in both cases.

Now, we shall prove our assertion about the modes. For u s 0, the function p_{a, u , n} is symmetric hence the mode is zero. For an ) 2 and

Ž . X Ž . Ž .

0-u - min par2, p y par2 , we have p_{a, u , n} 0 ) 0 by Theorem 3 ii ,

Ž . Ž U.

hence the mode is positive. For a, u, n g int EPD such that an F 2 and u F prn y par2, the number of zeros of p_{a, u , n}X is equal to zero on

Ž . Ž . Ž .

both y`, 0 , 0, ` by our argument in case ii , hence the mode is zero.

Ž . Ž U.

For a, u, n g int EPD such that an F 2 and u ) prn y par2, we

X _{Ž . Ž .}

q

have limxª 0 pa , u , n x s q` by Theorem 3 ii hence the mode is

posi-tive.

Ž .

Proof of Theorem 5. i Note that, under the conditions mentioned in

Ž .

part i of the theorem,

<_{1q e}yiuyip a r2 a_{y <F 1 q e<} iuyip a r2 a_{y ,}<

sin

Ž

n arg 1 q e

Ž

iuqip a r2ya

.

.

G sin

Ž

n arg 1 q e

Ž

yiuqip a r2ya

.

.

. Ž .

The proof follows from those inequalities and Eq. 22 .

Ž .ii It is easy to see that under the conditions mentioned in part iiŽ .

< yiuyip a r2 a_{< Ž Ž}

of the theorem, 1q e y decreases and sin n arg 1 q

iuqip a r2 a_{.. Ž .}

e y increases as a function ofu. The proof follows from Eq. 22 .

5. ASYMPTOTIC BEHAVIOUR AT INFINITY

Proof of Theorem 6. Fix ya - c - 0. Denote the boundary of the

 Ž . < < 4

region z: y n q 1r2 a - Re z - c, Im z - R by L . Applying_{n, R}

Cauchy’s residue theorem to the integral

f"

Ž .

z xzy1dz, x) 0,

E

a , u , n

we obtain f"

_{Ž .}

z xzy1_dz

E

a , u , n Ln, R cqiR y nq1r2Ž .aqiR _{" zy1} s

½

_H

q

_H

5

f_{a , u , n}

Ž .

z x dz cyiR cqiR Ž . cyiR y nq1r2ayiR _" zy1 q

½

_H

q

_H

5

f_{a , u , n}

Ž .

z x dz Ž . Ž . y nq1r2aqiR y nq1r2ayiR n " zy1 s 2p i

Ý

Reszsyqa

Ž

fa , u , n

Ž .

z x

.

.

Ž

24

.

qs1 Ž . Utilizing 3 , we have q y1 sin p qar2 " qu G ¨q q

Ž

.

Ž

.

Ž

.

" zy1 yqay1

Reszsyqa

Ž

fa , u , n

Ž .

z x

.

s _{2G n sin p qa G yqa G 1 q q} x .

Ž

.

Ž

.

Ž

.

Ž .

The bound 5 implies that the second and fourth integrals in the right Ž .

hand side of 24 ª 0 as R ª `. Thus, for x ) 0, we have n " zy1 pa , u , n

Ž

"x s

.

Ý

Reszsyqa

Ž

fa , u , n

Ž .

z x

.

qs1 1 y nq1r2Ž .aqi` _{" zy1} q

_H

f_{a , u , n}

Ž .

z x dz.

Ž

25

.

2p i y nq1r2Ž .ayi` Ž . Ž< <y1yna. Evidently, the integral in the right hand side of 25 is o x , which completes the proof.

6. SERIES REPRESENTATIONS OF p_{a, u , n}

Proof of Theorem 7. Note that if one of the conditions in the statement

Ž .

of the theorem is satisfied, then the Diophantine equation 13 has no

 4`  Ž .4`

solution. Thus the sets qqs1, a n q q qs0 which are the singularities of

q y _{ 4}

f_{a, u , n} and of f_{a , u , n} in z: Re z) 0 are disjoint. Hence, all singularities of

q q _{ 4}

f_{a, u , n} and of f_{a , u , n} in z: Re z) 0 are simple poles.

Ž . 

Fix 0- c - min an, 1 and let L be the boundary of the region z:_k < < 4

Re z) c, z - Q . The direction on the boundary is counterclockwise._k  < < 4

Denote by C_k and D_k the contours L_kl z: z s Q_k and L_k_ C ,_k

Ž Ž . Ž

respectively. The number Q belonging to the interval_k a n q k , a n q k ..

Ž Ž . Ž ..

Each of the intervals a n q k , a n q k q 1 contains at most two

 4` Ž .

points of the set qqs1. If it contains none, we set Qks n q k q 1r2 a. If it contains one, q say, we choose Q so that the distance from Q tok k k

Ž . Ž .

the nearest of three points q ,k a n q k , a n q k q 1 be not less than ar4. If it contains two, q and qk kq1 say, we choose Qks q q 1r2. Notek that the distance between Q and any pole of fk qa , u , n and of fya , u , n is not less than ar4 for any k s 1, 2, . . . .

Consider the integral

1 _zy1

" _{< <}

f

Ž .

z x dz.

E

a , u , n

2p i Lk

Applying Cauchy’s residue theorem, we obtain

1 _{" zy1} < < f

Ž .

z x dz

E

a , u , n 2p i Lk 1 _zy1 1 _zy1 " _{< <} " _{< <} s _{2p i}

H

fa , u , n

Ž .

z x dzq_{2p i}

H

fa , u , n

Ž .

z x dz Dk Ck k zy1 " _{< <} s

_Ý

Reszsa Žnqq.

_Ž

fa , u , n

Ž .

z x

_.

qs0 " _{< <}zy1 q

_Ý

Reszsq

_Ž

fa , u , n

Ž .

z x

_.

.

Ž

26

.

Ž . 1FqFa nqkq1r2 Ž . "

Utilizing the formula 3 for f_{a, u , n}, we obtain

" _{< <}zy1 Reszsa Žnqq.

_Ž

fa , u , n

Ž .

z x

_.

qq1 y1 sin n q q par2 " u G n q q

Ž

.

Ž

Ž

. Ž

.

.

Ž

.

_aŽnqq.y1 < < s x , G

Ž

n sin pa n q q G a n q q G 1 q q

.

Ž

Ž

.

.

Ž

Ž

.

.

Ž

.

" _{< <}zy1 Reszsq

_Ž

fa , u , n

Ž .

z x

_.

q

y1 sin p qr2 " u qra G qra

Ž

.

Ž

.

Ž

.

_qy1

< <

s x .

Ž

27

.

a G n sin p n y qra G q G 1 y n q qra

Ž

.

Ž

Ž

.

.

Ž .

Ž

.

To estimate the integral along C , we note that fork

zf B_a ` ` < <



4

[

D

z : zy q -ar4 j

D



z : zya n q q - ar4

Ž

.

4

qsy` qsy`

the following bounds are valid,

<sinp z G C exp p Im z<

Ž

< <

.

and < < sinp n y zra G C exp p Im z ra ,

Ž

.

Ž

.

where C is a positive constant not depending on z. Note that

sin

Ž

p zr2 " u zra F exp pr2 " ura Im z .

.

Ž

Ž

.

.

Moreover, for any M) 0 Stirling’s formula yields

< <



4

1rG z F C exp

Ž .

M

Ž

p Im z r2 y M Re z

.

when zg z: Re z G c , where CM does not depend on z. Finally, using Stirling’s formula once more, G zra

Ž

.

_ny1 < <



4

F C z1 when zg z: Re z G c . G 1 y

Ž

n q zra

.

 4 Hence in z: Re zG c, z f B_a zy1 ny1 " _{< < < < < < < <} f_{a , u , n}

Ž .

z x F A exp yp Im z r2 y M Re z q Re z log x zM

Ž

.

, where A_M is a constant not depending on z. In particular, choosing

Ms log D qpr2 where D is a positive constant, we obtain

zy1 ny1

" _{< < < < < <}

_w

_x

f_{a , u , n}

Ž .

z x F A exp yp z r2 z

Ž

.

, for xg 0, D .

Hence, as kª `, the integral along C tends to zero uniformly withk

w x

respect to xg 0, D . This proves the theorem.

Proof of Theorem 8. In virtue of Theorem 7, this theorem will be

Ž . proved if we show that both of the series in the right hand side of 12 converge absolutely for any x.

Ž .

If the condition iii in the statement of the theorem is satisfied, then

< Ž .< < Ž .<

obviously sinpa n q q and sin p n y qra are bounded from below by Ž .

a positive constant, whence both the series in 12 converge absolutely for any x.

Ž . Ž .

Since the proof in the cases of conditions i and ii is similar, we will prove only the former case.

Evidently, for any q) 1ra, there is an integer j such that_q a n q q y j s a k q lq rl y j - 1r2.

Ž

.

q

Ž

.

q

Since a is not a Liouville number, there exist an r G 2 such that

j l_q 1

a y G r.

kq lq

Ž

kq lq

.

Thus, for any q) 1ra, we have

1yr kq lq

Ž

.

F a n q q y j s a k q lq rl y j - 1r2.

Ž

.

q

Ž

.

q l Hence 2 _1yr sinpa n q q s sin p a n q q y j

Ž

.

_Ž

Ž

.

q

_.

G

Ž

kq lq

.

. l Ž .

Therefore, the first of the series in 12 converges absolutely for any x. Obviously, for any q) krl, there is an integer j such thatq

1 k jq 1 y y F .

Ž

28

.

a ql q 2 q Hence k jq 1 1 k jq 2 q F q y y - . ql q a a ql q a Thus 2 ql kq lj Fq . a

Note that kq lj G 2. Sinceq a f L, we have, for some integer r G 2,

ql _yr

a y G k q lj

_Ž

_q

_.

.

kq ljq

Ž . Ž .

Multiplying this inequality by kq lj r l_q a we obtain

1yr yr

kq ljq q q

_Ž

kq ljq

_.

2 2 l _1yr

y s n y q j Gq G 2

ž /

q .

Ž

29

.

l a a la a a

Ž . Ž .

The inequalities 28 and 29 yield

yr

q 2 2 l _1yr

1r2 )n y q j G_{q 2}

_{ž /}

q

for q) krl. Therefore, for such q, we have

yr

4 2 l _1yr

sinp n y qra s sin p n y qra q j

Ž

.

Ž

q

.

G _a2

ž /

_a q . Ž .

Thus the second of the series in 12 converges absolutely for any x. Ž .

Proof of Theorem 9. We shall construct a subset D of EPD which i is

Ž . Ž . Ž .

dense in EPD, ii is uncountable, and iii is such that, for a, u, n g D, Ž .

both the series in 12 diverge.  l4`

Let sn ns1 be a sequence of rapidly increasing integers defined by the equations

3sl

l l n

s s 1 q 2l,1 snq1s 1 q 2l

Ž

.

, ns 1, 2, . . . .

Ž

30

.

 4`

Denote by D the set of all 0 y 1-valued sequencesl dj js1 satisfying the conditions:

Ž .i d is allowed to be equal to 1 only if j g sj  n nl4`s1; Ž .ii infinitely manyd ’s equal to 1.j

 ` Ž .yj  4` 4

Let V s y: y s Ýl js1 jd 1 q 2l , dj js1g Dl and let L be thel

Ž .

subset of 0,` having the form t

yj

l s

_Ý

a 1j

Ž

q 2l

.

jsys

 4

for some s, tg N, a g 0, 1, . . . , 2l . Setj

Ek , ls



a g 0, 2 : a krl s x q y, x g L , y g V .

Ž

.

l l

4

Ž .

Evidently Ek, l is uncountable and dense in 0, 2 for any k, lg N. Set

Ds



Ž

a , u , n g EPD: n s krl g Q, a g E , par2 q u f L j Q .

.

k , l

Ž

.

4

It is easy to see that D is uncountable and dense in EPD.

Ž . Ž .

It suffices to prove that for any a, u, n g D the first of series in 12 diverges.

If a g E then there are integers m, i such that_{k, l}

m `

yj yj

a krl s

_Ý

a 1j

Ž

q 2l

.

q

_Ý

d 1 q 2lj

Ž

.

,

 4  4`  4`

where a_jg 0, 1, . . . , 2l and d_{j js1}g D . Denote by_l h_{n ns1} the subse- l4`

quence of sn ns1 such that

`



4

d s 1j iff jg hn ns1. Then for anyh ) m, we have_n

h m n yj yj 0-a krl y

ž

_Ý

a 1j

Ž

q 2l

.

q

_Ý

d 1 q 2lj

Ž

.

/

jsyi jsmq1 ` yj s

_Ý

d 1 q 2lj

Ž

.

jshnq1 yhnq 1q1 - 1 q 2l

Ž

.

. Ž .hn

Multiplying this inequality by 1q 2l , we see that there is an integer pn such that hn h yhn nq1q1 y 1r2Ž .hnq1 0-a k 1 q 2l rl y p - 1 q 2l

Ž

.

n

Ž

.

- 1 q 2l

Ž

.

, 31

Ž

.

Ž . for sufficiently large n. Consider the terms of the first series in 12 with

ŽŽ .hn _. qs q s 1 q 2l_n y 1 krl. Ž . From 31 we obtain hn sinpa krl q q

Ž

n

.

s sin

Ž

pa k 1 q 2l rl

Ž

.

.

hn s sin

_Ž

pa k 1 q 2l rl y p p

Ž

.

n

_.

Ž . y 1r2hnq 1 -p 1 q 2l

Ž

.

.

Ž

32

.

Sincepar2 q u is an irrational non-Liouville number, then as in the proof of the previous theorem there is an integer rG 2 such that

1yr

sin

Ž

Ž

n q qn

. Ž

par2 q u

.

.

G 2 k q lq

Ž

n

.

rl.

Ž

33

.

Hence, for sufficiently large n we have

aŽnqq .y1n < < sin

Ž

Ž

n q qn

. Ž

par2 q u G n q q x

.

.

Ž

n

.

sin

Ž

pa n q q

Ž

n

.

. Ž

G a n q q

Ž

n

.

.

G 1 q q

Ž

n

.

2 _{1yr a Žnqq .y1 Ž1r2.h} nq 1 n < < G _{p l}

Ž

kq lqn

.

x

Ž

1q 2l

.

Ž1qqn.2 a nqq2Ž n.2 = 1 q 2l

Ž

.

Ž

1q 2l

.

.

Ž

34

.

 4`  l4`

Since h_{n ns1} is a subsequence of s_{n ns1}, then

3hn 3

hnq1G 1 q 2l

Ž

.

s 1 q lq rk .

Ž

n

.

Ž .

Hence from 34 the series diverges.

We omit the proofs of Theorem 10 and Theorem 11, since they are similar to that of Theorem 7. The only difference is connected with double poles of f_{a, u , n}" and a more complicated form of residues.

ACKNOWLEDGMENT

The authors thank the anonymous referee for suggestions which improved very much the presentation of this paper.

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