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The Quantum Lobachevsky Space and the g-Bessel-Macdonald Functions
M.A.Olshanetsky1 ITEP, 117259, Moscow e-mail [email protected]
V.-B.K.Rogov2 MIIT, 101475, Moscow e-mail [email protected]
1 Classical case
Let L3 = 56r2\5,L2(C)be a homogeneous space of the second-order unimodular Hermitian positive definite matrices, which is a model of the classical Lobachevsky space. Let
9 = ( “ 5 ) > cx6-Pj = 1.
Then any i6L3 can be represented as
x= \ ( aa + j7 aj + jj \
99 \ pa+ 67 /?/? + 66 ) '
The Iwasawa decomposition
g = kb, g € SL2(C), k £ SU2, b £ AN, (1-2)
AN - Borel subgroup, allows us to define the horospherical coordinates on L3. If
h hz
then from (1.1)
6-1 0 h~l I ’
‘*‘=1 0%iA.-+W‘ I' (U)
The tripl (H + hh,z,z) is uniquely determined by x. It is called the horospherical coordinates of x. It follows from (1.1) and (1.3) that
H = aa + 77, Hz = a(3 + 7 6, zH = /?a + 6 7.
Let
*=(; “)• B-d D- c=(° ?)■ D=(”:
be the generators of the Lie algebra gl2 and d.A,d,B, dc and do = —Aa be the corresponding Lie operators
of right shift on L3. In the horospherical coordinates they take the form
dA = ^HdH - zdZ) dB = dz, dc = HzdH - z2dz + H~2dz. (1.4)
The cecond Casimir
n - -iA-r aD in the horospherical coordinates takes the form
Q — d~A + d2D + dgdc + dcdg
Q=1-H2dj1 + ~HdH + 2H-2dj2. (1.5)
Supported in part by RFFI-96-18046 grant
2Supportedin part by NIOKR MPS grant
Consider the eigenvalue problem
[\n+^Fv(z,H,z)='^Fv{z,H,z), v > 0. (1.6)
After the Fourier transform with respect the variables z and z we have the ordinary differential equation for the Fourier image of Fu(z, H, z)
(\и2^p + IhJh~ h~2~ss + H>s) = H’s)- (L7)
The solutions to equation (1.7) decreasing for H -> 0 are the functions
Ф„(«, H, s) = ■щ^-^Н~1Ки{2л/Istf-1)(ss)^(s, s), (1.8)
where Kv is the Bessel-Macdonold function, and ф„(Е, s) is determined uniqualy by <£„(s, tf, s). It is well-known fact that
—L—H-'KrpVteH-'Kss)*
T(u+l)
is the Fourier transform of the function
Pv(z,H,z) = (zHz + H-i)-v~1. (1-9)
After the inverse Fourier transform we obtain the solution to equation (1.7) in form
F„(z,H,z) = Pv(z,H,z)*f(z,z), (1-Ю)
where f(z, z) is the inverse Fourier image of <p(s, s).
Function (1.9) is called the Poisson kernel, and convolution (1.10) is called the Poisson integral.
2 Quantum Lobachevsky Space
Let Ag(SL2(C)),q Є (O', 1), be the algebra of functions on SZ,2(C) [2], which is defined as the factor algebra of the associate C-algebra with generators а, 3, j, 6 with an anti-involution * : Aq —+ Aq, (ab) = b* a* and the following relations
а в = q3a, a~f = 97Q. 06 = q60, 76 = q67, 0~> = 7 3,
a6 — q3y = 1. 6a — q~l 3~f — 1) 0a‘ = 9 la* 0 + q X(1 — q~)7 <5.
7 a" = qa“j, 6a* = a*6, 7/?* = /3*7,
63’ = q3’6 - q(l - q2)a*j, 6j*—q~1j*6. (2.1)
aa* — a*a + (1 — q1)7*7, 33* = 3”3 + (1 — q2)(&* & ~ a a) — (1 — l‘ ) 7 7i
77* =7*7, 6(5* = <5*6 - (1 - g2)7*7-
The rest commutative relations can be read off from the rule (ab)* = 6’a*. We cast the generators into the matrix form
a 0 \ » _ ( a* 7
- 1 7 6 ; ■ “ - v P' 6"
With the comultiplication A : Aq ^ Aq Aq
a P \ _ ( a 0 \ хгл ( a P
Al 7 6 ; - V 7 <5 V 7 5
the antipode 5 : .4? —*• Aq
s(° 1) = ( 6
7 6 1 \ -q'f a
and the counit e : Aq
Aq becomes a Hopf algebra. In fact it is a *-Hopf algebra since
(A(a))« = A (a*)
and
So*oSo* = id. (2.2)
We define the *-Hopf subalgebra Aq(SU2) by the generators
and the relations
Then
In a similar way
a*cac + 7c7c = 1, aca*c+q2 7c7c = 1,
7* 7c =7c7c, (*c7c = 97* «c, <*c7c = 97c<*c-
w>e = ( J J j . (2.4)
AW)={^=(J A-i)} (2.5)
/i/i* = h*h, hn = gn/i, hn* =
nn‘ = n*n + (1 — q2)((h* h)~2 — 1).
The Iwasawa decomposition in the quantum context takes the form [2]
lL>=UlcU>d, U) £ Aq(SLi2(C)), Ulc G Aq(SU2), Wd£Aq(ANq). (2-6)
Natural description of commutative relations (2.1) can be obtained from the construction of the
quantum double. It was implemented in [3],where .4,(SL2(C)) is described as a special quantum double of Aq(SU2), and (2.2) is derived by means of the corresponding iJ-matrix.
Definition 2.1 The quantum Lobachevsky space L3 is a *-subalgebra of Aq(SL2(C)) generated by the bilinear constituents
a*a + j*y a*/3 + j*S \ _ f p s
w ^ 1 /3*a + 6*7 /3*0+ 6*6 ) ~ { S* r )
Evidently, * acts as ■
* / \* * *
p — p: {S) = s , r — r.
We don’t need the explicit form of the commutative relations between p, s, s* and r - they can be derived from (2.1). .
Introduce a new generator 2 instead of n
n = hz.
Then due to (2.4), (2.5) and (2.7)
p = H = h’h = hh‘, s = Hz, s* = z*H, r = z*Hz + H~l. (2.8)
Consider now the complex associative algebra Uq(SL2(C)) with unit 1, generators A, B, C, D and the
relations
AD = DA=l, AB = qBA, BD = qDB, AC = q~1CA, CD = q~1DC,
A2 — D2
[8,0]=^-^-. (2.9)
In fact it is the Hopf algebra where
= A (D) = Z?0Z>,
MB) = -4 0 B + B (g) D, A(C) = A 0 C + C 0 D, (2.10)
£( C n) = (o 1 )> (2.11)
A A B)-( D -ri5 \
V c ^ y I -?C 4 J • (2.12)
There exists a non-degenerate dilinear form (u,a) : Uq x Aq ^ C such that (A(u),a06) = (v,ab), ('/0r.A •. =
(1 u,a) = e>t(n), (u, 1,4) = £u(u). (S(u), a) = (u, 5(a)).
It takes the form of the generators
(;?)>=('? ,-w. <*(; ih('7
(B>( ? f )) = rs IV (c, fa 8A) = (° 0
6 J \0 oj’ \ 7 ^^l 0 ;• (213)
Moreover, Lq(bL2(C)) is the *-Hopf algebra in duality, where the involution is defined by the pairing
(u*,a) = (u,(5"(a))-). (2.14)
The element
fi, = (g"1 +g)(-42 + ^>2)-4 , 1
2lrrZlfi +2 (BC + CB) (2.15)
is a Casimir element, since it commutes with any u 6 Uq(SL2(C))
The right action of u £ Uq(SL2(C)) on A is defined as [4] '
a u = (u 0 id)(A(a)). (2.16)
It is the algebra action:
a.(uv) = (a.u).v (2.17)
which satisfies the Leibnitz rule
(ab).u = J2(a-u})(b.u]) (2 18)
3
where A(u) = uj (g) u?. The left action is defined in the same way.
The right action on the generators takes the form "
° P) A - ( ql/2° q~1/2(] \ ( a P\ n ( 0 Q
- 6 )-A7 ,-<IH )■ ?J-C=(0 ,
“ f U’= ( P ° ) ( a P'\ D-( *"lla l'ni>
1 6 / V « 0 ) • I T « ) { ,-1/S7 q''H ,
We will! definenow the right action of U, (.91,(0) on LJ, which endows the latter with the struct,,,e
of the right -module. For any a 6 L3 define the normal ordering using (2.1)
*a+ -Y^Ckai,ka 2,* (2.20)
where a1 k(a2 k) are monoms derpending ov a* 3* -y* 6*(iy ft -v Thor, fV, ■ i . ,. q
will de denoted as (a).u, is defined as follows ( ’*7’ ^ ^ actlOD °n LI whlch
(a)-u = ^2 ckai,k(a2,k-u). (2 21)
The generators H, z are expressed by generators a,... ,5* as
z = a!-ll3+J2(-l)kq-2k(y*)k + 1yk*-2, y = 7a-1. k-0 '
(2.23)
TBM^:i27Z:rTTusins(219X (2 21) • (2-23) wc c" *fi”the ** “«» ^
w(m, r, n).A = q~n+^w(m, r, n),
i 1 2n
u>(m, r, n).B = ^ n _ ^
r- ")c=-1, r _ 2, „) _ r_ n+1}i
w(m,r,n).D = qn-tw(m,r,n). (2 24^
The second Casimir (2.15) acts on monom w(m, r, n) as
w(m, r, n).Q = (]l~r (I--1---] T. 7,^ , r-i (1 - rm)(l - q2n)
' V l~q2 J u^'n’T’n) + <J -------------_ q2y----------v(m-l,r-2,n- 1). (2.25)
Remark 2.1
3^3,,! = ^, U;n/,Z> =
and
1 l\nnB~dB, lim C = dc,
i-o 9-i-o ’
lirn Q, = + I.
?-i-o q 2 4
3 Modified g-Bessel Functions
We »,,d the fundamental formula., from the theory of the basrc hype,geometrical series. For any
(a, q)n = <* % for n = 0
' I (1 ~ a)(l - o-q) . . . (1 - aqn~1) for n> 1, ’
(a> q)oo = nlm^(a,9)rlI (ai! . ..,ak,q)x = (all9)oo . . . (ak, q)^. (3.1)
The q — I-function is defined in following way
r*M=(F^(1
The basic hypergeometrical funcyion is
OC / ' \ /
....‘..........W
The 9-exponents are
= 1*1 <i,
71=0
x
Fl<1> (3-3)
OO »(n-l)
Eq(x) = y .
^ (?■?)" (3'4)
Ihe g-exponents can be represented by forms:
e?(2r)=(^U’ E*(x) = (-*,q)oo. (35)
There are two typies of g-trigonometric functions
cos? X = -[eq(ix) + eq(-ix)l sin? ar = ±-.[eq(ix) - e, (-,*)], (;] g)
Cos? i + £,(_;*)], sin? a: = ~[Eq(lx) - Eq{-ix)]. (3.7)
Let consider the complete elliptic integrals
r*/2
K(I-) =
Jo
da rir/2
,----====, K'm= i_______________“
and let
lng=-ZS2 * K(*) • (3.8)
Then [6](5.3.6.1)
OO
Q„ = (1 - 7) y ___________L________--1_Vau r21nrHj ,
+ ~ 7T ( ) f-;------------K'(*)]. (3.9)
^ llCIC d fl U ~z \f \ — ~ SIM ^ 7/ — ( ^ d'.i /1 • , .
_ ’ ^ ^ “ 1S 16 JaCobl elliptic function). U It = 0 then
o _ 0 and dn u - 1. It followes from (3.8) and (3.9) for any v
lim Qu = — 2
In []j the g-Bessel function were determined as
= T^(s/2)" -^£1)
Lhe^oliovving 1)- We **“ th'"'»d'fed ^ *W) » .1...
#\s-,q) = e-iT4^(e“fs;?)i j = l,2 (3 10)
We will consider below the functions
41]((1 - g2kr) = --------___-V" (l-g2)2ns^+2n !
IV('y + 1) “ ('72,g2)n(g2,/+21g2)ri2I'+2'>’ '*'< n-^ (3.11)
/(2)((1_92)s;(?2)= _____1_____^ q^+n){l_q2fns,+,n
r?2(i/ + 1) (^(5^+;, g2)n2^+2n • (3-12)
If |g| < 1, the series (3.12) converges for all s =£ 0 Therefore /(2Vn „2^ 2\ • , , ,
outside of 2 — 0. J-nereiore Lu ((1-5 )s, q ) is holomorphic function
Remark 3.1
?hrn o^;((1-?2)5;92) =/„(s), i=li2
The function /^((l - q2)s\q2) is the meromorphic function outside of 2 = 0 with the ordinary poles ir the points s 7^ ± , r — 0,1,...
Remark 3.2 If q —* 1 — 0 the all poles of /£^((1 — q2)s] q2)
, 2g-r
-----2, r = 0,l,...
1 — qz
go to infinity along the real axis.
We have from (3.11) immediately Proposition 3.1 The function /^((l - q2)s;q2) satisfies the following relations
i'J) = «-'/'V+itd - ?2)«; r),
23.,
1 + q
and the difference equation
1 - q2)s-,q2) = -q2)s]q2),
[1 - (^^)2?“2s2]/(r1s) - (?" + q~lf(s) + f(qs) = 0 (3.13)
Analogy we have from (3.12)
Proposition 3.2 The function li2\( 1 - q2)s;q2) satisfies the relations
- »W) = !%,((! ~ ?!W).
9 f)
i_TTs‘'/P)((1 - (i2)s^q2) = 9“'/+1sv-14-i((1 - 92)9a’;?2).
+ q
and the difference equation
' Л'Г1*) - (qV + ?-")/(*) + [1 ~ (Ц^)2^]
It is easy to show that the functions (3.11) and (3.12) are connected by correlations:
- q2)s-,<f ) = сг’(^-у^-г2)^2)((1 - r>;?2) (3.15)
/^((1 - q2) = ^(-Ц^52)/Г°((1 - ?2)s; g2) (3.16)
4 The g-Bessel-Macdonald Function
Unfortunately the function /^((l - q2)s;q2) is determined by power series (3.11) in domain js[ < ~~y
only while we need a representation of this function as series 011 the whole complex plane. But the
following proprsition takes place '
Proposition 4.1 The function lil\( 1 - q2)s\q2) for s ф 0 can be represented as
4])(.( 1 ~ q')s',q-) = ^-[eq{~^z)<bv{s) + іе^еі{-1-^-г)Ф1,(-з)]1 (4.1)
where ■
Ф„(в) = ■2$l{qV + ll2,q-v + ll2\-q-,q, ^ 2, ), (4.2)
and
M-1)^ (4-3)
The coefficients а„ (4.3) satisfy the recurrent relation
du+i = avq~v~1!'1
and the condition
Cl у CL _ у —
q-» + V 2
2r?a(i/)r?2( 1 — u)smi/Tr
Proposition 4.2 The function I{u2\(\ _ g2)s-(72' f , „
^ V ^ j ? ) jor 6 f 0 can be represented by
In the classical analyses the Bessel-Macdonald function is defined as
AV(5)=2^f/-^)-/‘'(5)] (4.5)
for V ± n, and if V = n by the limit for u ~ n jn (4 V ,
"it,on m ”ch wa) that •*» »f
Definition 4.1 Tkc ,-B'ssel-.\I.clon.U f,,nc„„, (q.Bi,F) are lefineJ „ '
(i e}
with av (4.3), j = \, 2. .
the limit forC^C^C(4S6)thlS def,mt'0n Sh0uld be adJ^ted for the integer values of the index * = „ hv I! follows from (4.1), (4.4) and (1.6) ’
к(1)((л 2\ o, q~v2 + ll2 1 _ 2
4(1 -q )s: q~) = _i---------------_e / i - q
9 Г7Гr, /Te?V ГГ-
V^TT~2~5)ф"(~5). (4.7)
^ 2 4 *)• (4.8)
It is casu to prove using (4.7) and (4.8) the following
Proposition 4.3 4-BMF KW4\ - -,-M ■ • -2* • , ,
" ' 4 7 кч a 0'0rnorphic fvnrtion in domain Res > ,2у.
Proposition 4.4 q-DMF K[2)m _ ^ , *
" (( 9 >■? ) ;л- a holomorphic function in the domain s ф 0.
T he next propositions take place
Proposition 4.5 The function K(V)(n _ ,,2\,. 2x , • ,
" ^ 1 ) satisfies the following relations
2ds ,
l+qS K {{l~q2>’(l2) = ~sU~1I<il}1(^--q2)S:q2),
i + ~ ?W) = -s—iA'0)i((i _
о»г/ /Л<? difference equation (3.13).
Proposition 4.6 The function K^2)(n ^
/ /z0n Ky ((1 _ r)s. q } saUsfies ihe foUow.ng rclatwns
20 S
- Q2)s: q2) = -q-^s^K^l - q2)s’, q2),
2ds _
1 + / ^ Я2) = -r^s-^Ki^d 1 - 92),; ?2)
and the difference equation (З.Ц).
Proposition 4.7 For any 1/ the functions /^((l-g2)s;?2) and A'<1}((1 - q2)s; q2) form a fundamental system of solutions to the equation (3.13).
Proposition 4.8 For any v the functions /<2)((1-- g2).s;g2) and Af°((l - g2)s; g2) /orm a fundamental system of solutions to the equation (3.14)-
Remark 4.1 o
lim A'^((l —q )s',q ) = A'„(s), J = 1,2.
g — 1-0
Remark 4.2 // g —► 1 - 0 the representations (4-1), (4-4), (4-7) an& (4-8) g^ve us the well-known asymptotic decompositions for the functions Iv(s) and K„(s) respectively [7].
5 The Jackson Integral Representation of the Modified g-Bessel Functions and g-Bessel-Macdonald Functions
Jackson g-integral is determined as the map an algebra of functions of one variable into a set of the number serieses ^
[ /(*)cf?* = (i-g)X>H/(gm) + /(-9m)].
J — I m — 0
co
rCO _
I f(x)dqx = (l-q) y gm/(<?m)>
J 0 7X1 — — CO
oo
/oc „_
/(.r)dgx- = (l-g)^gm[/(gm) + /(-gm)]-
-oo -CO
Define the difference operator ^
(5'1}
The following formulas of the g-integration by parts are \alid
I rl
J &(x)dMx)dqx = 0(\)u( l) ~ 6(-l)u(--l) - J ^o(r)uiqx)dqx: (5.2)
A <t>(x)dM*)dt* = Iim^^(?-mMrm)-^(9T"M7m)]-^ dxd>(x)iLiqx)dqx. (5.3)
Jq m co 0 ^
P *(*)«*)</,* = JLm [o(g-m)^-(g-"1) + - J_x ds<t>(x)v{qx)dqx. (5.4)
J — OC
The last- two expressions imply the regularizations of the improper integrals.
Let z and s be noncommuting elements and
(5.5)
r.s gsr.
Consider the function
/(*) = ^a,.zr.
r
T/ie rule of q-integration in the noncornmutative case
J f(zs)dqs = j yar(zs)rdqs = j ytarq-r-L^izrsTdqs, j dqzf{zs) = j d^arizsy = J dqzy/arq~A^JzrSr .
Define the following transformation ift for functions / depending on the noncornmutative variables s and z (5.5). If we have function which has the form (5.6) and all monoms are order we will write
f(Z8) = ytar(zs)r -
Definition 5.1 The function /(г) absolutely q-integrable if the series
ОС
E ?m/(?m)
m~~ со
converges absolutely.
It means, in particular, that
| = 0
It follows from (3.2) - (3.4)
Proposition 5.1
TP ~~ Q \ i yl~Q^
£,(-T-«) = te,(58)
There is a g-analog of classical binomial formula [5]
00 ( \
(1 _ - уМаЬ * T(a + k)
’ -Wt= вд ■■ ''К1-
{<1агл)сю \^(qa,q)k к
-Ет^тгг- N<1-
(z,q)0о ^ (9, g)*
We need in two generalizations of this g-binom
r(a,b,z,q) = (g5’g)°°
(^,?)oo (5-9)
- Ii(a, b, 7 z, q2) = (^_iS_)oo ^
1 ^2,?2)00‘ (5-10)
The function (5.9) satisfies the difference equation
4br(a, b, z, q) - ar(a, b, 52, 9)] = r(a, ^ ^ ^ ?) ^
The function (5.10) satisfies the difference equation
*-№R(a, b, 7, z. g2) - ^ b< 7. ^ g2)] = ^ ^ ?2) _ ^ ^ ^ ^ ^
Lemma s., //W < W c„ „ rlprcsenh, as * „„„ ,/||e ^ fin*,..
= —L_ f LlhzizMh^sht
(b~,q)oc (?>?)oc^ (?,g)*(i - ^69*) ' (5-13)
T/ie series (5.13) converges absolutely for any z ± * = 0,1
Remark 5.1 //0<|a|<|6| Men
(azi g)oo _ (a/6, g)^ (b/aq, q)k(a/b)k
(bz>q)00 (9,9)00 fr/0(q,q)k(i - zbqk)' (5 H)
If a = 0 then
1 1 00 ( i\k k(k + 1'>
77-----— = —-______\" (-1) g 2
(*‘=9)00 f^0(q,q)k(l - zbqk)' (5.15)
Assume that a = e92“,6 = e(,2/3 , , • ,
’ ^ ’ ±1 in (o.lO). Then we have from (5.14)
Corollary 5.1
Remark 5.2 /Is it follows from [5](1.3.2)
(eq2az2, q2)^ k Jpk {q2(-a~P\q2)k 2t (5 17l
(eg2/322,g2)oo ^ 9 (<Z2,92)* ' '
which converges in the domain |z| < q
It follows from Lemma 5.1 that if 7 = 0 (5.16) is the meromorphic function with the ordinary poles
2 = ±y^q-f3~k, k = 0, 1, ..and hence it is the analitic continuation of (5.17).
Corollary 5.2 For an arbitrary real s / 0
lim 1 = 0.
m—* co z
Corollary 5.3 For real s ^ 0 and integer m
, ,1 -?2 -m , , ^(r1)^?)
IW —!
Corollary 5.4 If a > P + I and real 2 / 0, then
(~q2az2,q2)00 < Cg„a
(~q^z2,q2)00 - l + z¥r
Remark 5.3 lei a = eg2a,6 = eg23 in (5.10) and (5.12). Then if q -* 1 - 0 iAe difference equation (5.12) takes the form of the differential equation
z(l -ez7)R’(z) - [7 + c(2a - 23 - 7)z2}R(z) = 0 (5.18)
with solution
R(z) = C z^ {\ ez2) .
Proposition 5.2 Modified q-Besscl function (q-MBF) l[l) for v > 0 can be represented as the q-integral
w-.w>=<519’
Proof. Consider the g-integral
ss>>(,)=(*•»)
where si'V) is such function that it rs absolutely convergent. Require that -^5L1 (s)(.s/2)1' satisfies (3.13).
Then 5(11)(s) satisfies the equation
' S[l\<-rls) - Si!)(s) - q2l'[Sl1l\s) - S(7(qs)) = (l^Sl)2q~2s[l\q^s)s2. (5.21)
Substituting (5.20) in (5.21), using the rule of q-integration, (5.8) and (5.2), we come to the difference equation for flX\z)
q2^lz2[fil\z) - q~2u+lfi^iqz)] = /<1}(z) - (5-22)
It coincides with (5.12) for a = q2,b = g2"+1,7 = 0> an<^ hence
f(D(z) - (g2z2.g2k_ ■ (5.23)
h (92, + l22i52)oo- V '
5|1-\s)(a72)‘/ is a solution to (3.13) and therefore it can be represented as (see Proposition 4.7) sf = ^41}((l - q2)s; q2) + BAf)((l - g2)S; g2).
Multiplying the both sides on (s/2)1' and putting s = 0 from (3.11) ( (4.6) we obtain 5 = 0. Multiplying again on (s/2)-" and assuming s = 0 we come to
Ld’zjt/,(z) = AT^r.
q'y1'" + !)
It follows from [5] (1.11.7)
2
i + - i + 9
and we come to (5.19).B At the same way we can prove the following
A = j—Be(u + 1/2, l/2)I>(j/ + 1) = + l/2)I>(l/2).
Proposition 5.3 The q-MBF li2\( 1 — q2)s; q2) for v > 0 has the following q-integral representation
4 )((1 - 9 )«. 9 ) = 2r^(i/ + 1/2)rffa(l/2) *
X J\dq Z (^--0) . 0; g- • (5-24)
Remark 5.4 If q —► 1 — 0 the equation (5.22) takes the form of the differential equation (see Remark 5.3)
{\-z2)f,u{z) + (2v-\)zfv(z) = Q.
The solution to this equation is
Mz) = C\l-z2y-1/\ which leads to the classical integral representation of Modified Bessel function [7] (7.12.10)
/ (s) = ______(,s/2)'/___ f1 (i _
A) T(v+l/2)T(l/2)J_l[ ■
Proposition 5.4 The q-BMF A’£^((l — g2).s;g2) for v > 0 can be represented by the q-integral
<r"a+1/2IV(* + l/2)r,3(l/2)
q = -------------------------------------------------------\/-------x
4Qi, v a_
-V
„ r , {-q2z\q2U ^.i-92 v /ow ,-0.,
/ g2)TC 2~-S)(V2) , (o.2o)
' —cc
where Q„ is defined by (3.9).
Proposition 5.5 The q-BMF A'|f ^((1 — g2)s; g2) for v > 3/2 can be represented by the q-integral
Ki„ , , = ,-*T,.(. + №|i;2)n7
4Ql/2 V a--"
ah
X
/•oo / 2i/ + l 2 2\ i 2
x I dqz—^—— o^i(-;0;g,i—y— zs){s/2) v, (5.26)
> — OO
where Qi/-j is defined by (3 9).
Remark 5.5 J/g —> 1—0 the representations (5.25) and (5.26) give us the classical integral representation of Bessel-Macdonald function (the Fourier integral) [7] (7.12.27)
References
[1] Podles P. and Woronowicz Sd. Quantum deformation of Lorentz group, Commun. Math.Phys., v.130 (1990) 381
[21 Jurco B. and Slovachek P. Quantum Dressing Orbits on CompactGroups, Commun.Math.Phys., v.152 (1993) 97-126
[3] Koornwinder T. Askey-Wilson Polynomials as Zonal Spherical Functions on the SU(2) Quantum Group, Report AM-R9013 (1990)
[4] Jackson F.II. The application of basic numbers to Bessel's and Legendre’s functions, Proc. London math. Soc. (2) 2 (1905) 192-220
[5] Gasper G. and Rahman M. Basic Hypergeometric Series, Cambridge. Cambridge University Press, (1990)
[6] Brychkov A., Prudnikov Yu. and Marychev 0. Integrals and Series, v. 1, Nauka, Moscow (1986)
[7] Bateman H. and Erdlyi A. Higher transcendental functions, v. 2 Me Graw-Hill Book Company.(1966)
