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q‐exponential and q‐gamma functions. II. q‐gamma functionsa)D. S. McAnally Citation: Journal of Mathematical Physics 36, 574 (1995); doi: 10.1063/1.531323 View online: http://dx.doi.org/10.1063/1.531323 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/36/1?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Functional Inequalities for Incomplete q‐Gamma Function AIP Conf. Proc. 1389, 1731 (2011); 10.1063/1.3636944 Integral representation of Dirac distribution using the Tsallis q-exponential function J. Math. Phys. 51, 123509 (2010); 10.1063/1.3520533 q‐exponential and q‐gamma functions. I. q‐exponential functionsa) J. Math. Phys. 36, 546 (1995); 10.1063/1.531322 The q‐gamma, q‐beta functions, and q‐multiplication formula J. Math. Phys. 35, 4268 (1994); 10.1063/1.530852 The exponential function—Part II Phys. Teach. 14, 485 (1976); 10.1119/1.2339464

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q-exponential and q-gamma functions. II. q-gamma functionsa)

D. S. McAnally Department of Mathematics, The University of Queensland, Brisbane, Queensland, Australia 4072

(Received 13 April 1994; accepted for publication 17 June 1994)

For 141 # 1, the integral definition of the gamma function in terms of the exponential function is generalized to a definition of a q-gamma function, or a family of q-gamma functions, in the standard and symmetric cases, using the q-exponential functions and their asymptotic behavior as z --+ ~0 on geometric sequences {zLC’ = cJ”~}~ Ez with common ratio q. The properties of the q-gamma functions (poles, zeros, product expansions) are also determined. 0 1995 American Institute of Physics.

I. INTRODUCTION

In the first article’ (henceforth referred to as I), q-exponential functions were defined, and their properties explored. Just as q-exponential functions have assumed new importance with the introduction of quantum groups, so also deformations of the gamma function are important in that new area of interest2-4 as well as in the more traditional area of q analysis. The definition of generalized gamma functions is by no means straightforward, as will be seen in Sets. III and IV, but requires the introduction of preliminary functions which are defined in terms of q integrals of q-exponential functions.

The important functional relation for the gamma function

T(t+ l)=zT(z) (1.1)

can be derived from its integral definition in terms of the exponential function. Similarly, the q-deformed version of the relation between factorials and integrals of the exponential function4 can be generalized to a definition of q-gamma functions in terms of convergent q integrals of q-exponential functions, and these q-gamma functions can be shown to obey a similar relation to Eq. (1.1).

In Sec. II, the definitions of q differentiation and the q-exponential functions (standard and symmetric), and the asymptotic properties (derived in I) of the q-exponential functions, are re- viewed briefly.

Section III opens with a review of the gamma function. This is then generalized in the standard case to a standard q integral on an appropriate geometric sequence of points and involving the standard q-exponential function. These deformed gamma functions are called q-quasigamma functions as the functional relation which they satisfy is not yet the form desired for q-gamma functions. For O-C lq[< 1, there is a parameter attached which depends on the geometric sequence taken in the q integral. The properties of the q-quasigamma function with O< [ql< 1 are then investigated for different values of this parameter. It is found to be an analytic function whose definition can be extended to the entire complex plane (as in the case of the gamma function). A product formula is found, and from this formula, it can be seen that the parameter enters into the functional form nontrivially. Parametric q-gamma functions satisfying the desired functional equation, are then derived both for 0 < 1 q I < 1 and for I ql > 1.

‘)Dedicated to the memory of Dr. Ansgar Schnizer (1962-1993).

574 J. Math. Phys. 36 (l), January 1995 0022-2488/95/36(1)/574/22/$6.00

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D. S. McAnally: q-gamma functions 575

For jql> 1, the q-quasigamma function is not parametrized as the only geometric sequence of interest is that containing the zeros of the q-exponential function. This is reflected in the notation adopted for the q-quasigamma function, with I q I > 1, whose properties are then investigated. Just as in the case of 0 < I q/c 1, it is found to be an analytic function and can be extended to the entire complex plane. A product formula is found. Nonparametric q-gamma functions satisfying the desired functional equation, are then derived both for 0 < I q I< 1 and for I q I > 1.

In Sec. IV, the gamma function is then generalized in the symmetric case by taking a symmetric q integral on an appropriate geometric sequence of points and involving the symmetric q-exponential function. Because of the long range behavior of the symmetric q-exponential function, only one sequence is appropriate, and because of the nature of symmetric q integration, only every second point is used in the q integral, so that there are two q-gamma functions, and the functional relations for them intertwine them. Their properties are investigated, and they are found to be analytic functions, and can be extended to the entire complex plane. Their sum and difference are studied as these are not intertwined by the functional equations. It is found that the relation between the gamma function and the trigonometric function sin( nz) can be generalized in the case of the sum and difference to an elliptic function (and a constant in one case). Product formulas for the four q-gamma functions (the two pairs) are found, and the previously mentioned constant is evaluated.

II. q DIFFERENTIATION AND @EXPONENTIAL FUNCTIONS

In standard q analysis, differentiation is deformed to the operation defined by

dqff(z) f(w) -f(z) -= d,z (4--l)z . (2.1)

If q EC, q # 0, 141 # 1, then there is a unique function exp,(z) (the standard q-exponential function) satisfying the following conditions:

dq eJ:(z) = exp,( z), 4

exp, is regular at z = 0, (2.2b)

exp,(O) = 1. (2.2c)

The power series expansion for exp, is given by

(2.3)

where (O),!= 1, (n),!=II;=,(k) n 3 1. If 0 < /ql< 1, the power series has radius of convergence l// 1 - q I. If jq/> 1, the po’der series has infinite radius of convergence, and exp, is an entire function. The product expansion for the standard q-exponential is given by

1 expq(z)= II;&1 -( 1 -q)@z)

for O<lql<l, and

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576 D. S. McAnally: q-gamma functions

exp,tz)= I-I I+ .II, ( Yzi

(2.5)

if Iql>l. If 0 < /q I< 1, then exp, is a meromorphic function with no zeros, and with simple poles at

4 --R

z=l-q, n=0,1,2 ,... .

If 1 q I> 1, the entire function exp, has simple zeros at

-4” z= 4-1’ n= 1,2,3 ,... . (2.7)

IfO<lql<l, 5+0,5 $ qZ/(l-q)

exp,(qFNb)-( - 1)‘vq(1’2)N(N+1)( 1 -q)-N{-Nf,(c) + 0 as N -+ ~0 (2.8)

[this is Eq. (3.18) in I] where f4 is as defined in Eq. (3.12) in I, and the decay is faster than the growth of any polynomial. That is to say, if p is a polynomial, then p(q-Nl)exp,(q-NJ) --t 0 as N + 00.

IfIql>l, 5 + 0, 5 6 -qZk-l)

exp,(qN5)--q (1’2)N(N-1)(q-l)NlNfq(J) -+ 00 as N -+ to (2.9)

[this is Eq. (3.22) in I] and the growth is faster than the growth of any polynomial. In symmetric q analysis, for example, for applications in physics,4-6 differentiation is de-

formed to the operation defined by

474 f(w) --J-(4- ‘4 -= d(w) (q-012 ’ (2.10)

which is invariant under the interchange q H q- ’ . If qEC, q f 0, 141 # 1, then there is a unique function E,(z) (the symmetric q-exponential function) satisfying the following conditions:

(2.11a)

E, is regular at z = 0, (2.11b)

E,(O)= 1. (2.11c)

This function is symmetric under the interchange q ++ q- ’ given by

. The power series expansion for E, is

(2.12)

where [ 0] 4! = 1, [n] 4! = II;= i [k] Q , IZ 2 1. The power series has infinite radius of convergence, so that E, is an entire function for all q # 0, [q I # 1.

If 0<1q/<l, then


On: Tue, 18 Oct 2016 06:17:31

D. S. f&Anally: q-gamma functions 577

-8 E, 1-q2 = ( i zpo q(‘/2)(n-2k-I)(n-2R-2)n[;a=k+l(1 -q2’)

c;=-, qzm2+, (2.13)

[this is Eq. (3.26) in I] and if 5 # 0, 5 $ qz, then there is a nonzero complex number A(l) such that

E -cN5

( 2) q l-q -(- ~)NA(s)~-(“~)N(N+~)~N as N ~ m

[this is Eq. (3.28) in I] so that

Eq -cN5 ( i l-q2 + m as N+ m,

(2.14)

(2.15)

III. 9 DEFORMATIONS OF THE GAMMA FUNCTION: STANDARD CASE

In ordinary complex analysis, the gamma function r(z) is defined by

I

m

T(z)= xzpl exp( -x)dx. 0

(3.1)

This integral is defined for all z with R(z) >O, so that I? is analytic on the open right half plane. The gamma function satisfies the functional equation

rtz+ i)=zryz) (3.2)

if !.X(z)>O. Also, r(l)=l, so that r(n)=(n-l)! if rz~Z, n>O. The functional equation (3.2) can be used to continue l?(z) analytically to the entire complex plane. Thus defined, r(z) is a meromorphic function with no zeros, and with simple poles at z = -n, n = 0,1,2,. . . The residue of T(z) at z = - n is ( - 1 )“ln ! . The gamma function also satisfies the functional equation

rwv -d= &* 7. E c.

The integral definition of the gamma function can be deformed to a q-integral expression. Firstly, note that for 0 < 1 q I < 1

expq(qeNl) + 0 as N -+ ~0 if 5 # 0, 5 $ & (3.4)

by JZq. (2.8). The gamma function can then be deformed to a single parameter q-quasigamma function by setting

f,(z;l)= lim I

q-Ni M,N+m q”5

exp((z - 1 Ilog x)exp,( -x)dqx (3.5)

for 5 Z 0, 5 $ - qz/< 1 - q), and specified choices of log q and log 5. Then

M-l

f,(z;c)= lim C M,N+m

(1 -q)qm~qm(z-l)~z-l exp,( -q”l) m=-N


On: Tue, 18 Oct 2016 06:17:31


= C (1 -4)PlZ exp,(-qm5), m=-02

(3.6)

which converges if and only if %(z log q)<O, i.e., lqzl < 1. [Note that the integral converges for all z as N A m, but as M -+ ~0 if and only if R(z log q)<O.] The dependence of r,(z;l) on log [is relatively unimportant since

From Eq. (2.4)

f,(z;J exp(27ri))=exp(2rriz)fq(z;5). (3.7)

1 1 ew,( - qm5) =

~;=,(l+ t l-4)4 “‘“5) = IG,(1+ ( 1 -q)qR!g (3.8)

and so

f,(z;l)= 5

(1 -4h?mZ5Z

m=-io ~;=,(l+u-dq”l)’

(3.9)

It is reasonably easy to verify from Eq. (3.9) that

fq(z;qNl)=f’q(z;l), if NEZ. (3.10)

Here, log(qNc) has been taken to equal N log q+log 5. Use of q integration by parts [or direct substitution of { [ 1 + ( 1 - q)qm[]/( 1 - q)} - l/( 1 - q)

for the extra factor q”l in any given term in Eq. (3.9)], gives

f,(z+ 1;5~=q-~~z~q-lfq(Z;5)=-~-Z)q~q~Z;S~. (3.11)

If z+ (2MmYlog q) is substituted for z in Eq. (3.9), then

=i(q(~;~)~2M”i’10gq, if ME Z.

Let

Sz,(z;S)=f,tz;5)log 5+ 5 (1 -4)wmz5z 1% 4 m=-oc ~,“=,(l+(b?w5)

(3.12)

(3.13)

then the series for fi,(z;l) conyerges absolutely for z such that R(z log q)<O, Since fi,(z;l) is the term-by-term derivative of I’,{z;l) with respect to z, and the series for a,(~;$) converges absolutely if !R(z log q)<O, then r,(z;[) is differentiable for R(z log q)<O, and n,(z; 5) is its derivative.

The functional equation (3.11) for f,( . ;[) can be used to extend the function analytically to the entire complex plane. Since q-‘(z),-1 = -( - z)~ has simple zeros at 2MwiAog q, M EZ, then l?‘,(z;l) is a meromorphic function with simple poles at

2Mrri z=-N+-

1% 4 ’ M,NsZ, NaO. (3.14)

Since


On: Tue, 18 Oct 2016 06:17:31


f,( l;{)= lim I

q-N5 exp,( -x)d,x= M,N-m q”i

lim [ -exp,( -x)&F M,N+m

=M~~_m(expq(-qM5)-expq( -qwN5))= 1

then

pq( 1+ z;() =52Mwil’ogq.

The residue of f,(z;<) at z=-N+(2Mri/logq) is

@es r’q(z;5)),=-N+(ZM~~llogq)=

(_ 1 )Ncq _ 1)62Mrillog q

W),! log q *

(3.15)

(3.16)

(3.17)

The proof is by induction on N, using Eq. (3.11). Since

,-. nlo_ (l;;;m*)qnl)=fi ~~+~~-~~~"+z5~~~+C~"+'-Z~~~-~~51~~~-~"+'~ n-m n-o (1+(1-4)4"5)(1+[q"+'/tl-q)51)(1-q"+z)

(3.18)

for 0 < [ql< 1, R(z log q)<O (this is proven in Appendix A), then

~q(z;5)=(1 -4)CZfi 1

l+( 1 -q)q”+zc 1 +[qn+‘-V( 1 -q)J] 1 -qn+l

n=. 1+t1--4w5 - ~+w+'~w4>51 .-

l-q"+z 1 =ez-lfi

[

l+( 1 -q)qnfZc 1 +[q”+‘-zl( 1 -q)l] 1 -qn+’

n=O 1 +t1-4w+Y .-

1+[4"/(1-4)fI 1 -g+z 1 (3.19)

and so tq(z; 5) has simple zeros at

z=N+ Wf+l)ri logtl-4) log 5 M NEZ 1% 4 - log 4

-- log q’ ’ * (3.20)

Note that these points are distinct from the poles because 5 $ - qz/( 1 - 4). If O<]q(Cl, and l# 0, 5 $ -q’/(l-q), let rq(z;lJ=q(1’2)z(z-‘)fq(z;~), then

rq(z+i;5)=tz),rqtz;5), r,ux)=i (3.21)

so that r,(z;l) can be regarded as a q-gamma function. In this case

2Mri ~;;i 2Mriz- E

1% 4 rq(Z;gpfwi@q. (3.22)

If Iql>l, and 5 f 0, 5 $ -qz/(q- l), let r,(z;c) = q'-zI=,-1(2;5>, then

rq(z+1;5)=(z),rq(z;5), r,ua=i (3.23)

SO that T,(z; 5) cm also be regarded as a q-gamma function. Note also from Eq. (3.12)


On: Tue, 18 Oct 2016 06:17:31


r,i z+ z;c) =rq(z;5)52Mvi’10gq.

For [qj> 1, the expression

5 exp((z- 1)log x)expq( -X)dqX

(3.24)

(3.25)

is only defined for 5 E qz/(q - 1 ), because

exp,(-qN5) + 00 as N -+ 03, if 5 # 0, 5 $ q’/(q-1) (3.26)

by Eq. (2.9). For specified choices of log q and log( l-q-‘) [e.g., the principal value of log( 1 - q- ‘)I, the q-quasigamma functions, which at present apparently depend on IZ, are given by

f’( z; -&) =M!;~mf~~~~~:I’: exp((z- lbg x)evq(-xMqX

cc q(m- l)z = c (4-l) (l-q-l)z

m=-02 (3.27)

which converges if and only if %(z log q)>O, i.e., lqzl> 1. [Note that integral converges for all z as N --+ 00, but as M + 00 if and only if R(z log q)>O.]

Note that there is no dependence on n (so that the q-quasigamma function is unique), so ib(z;q”/(q- 1)) can be denoted by I‘,(z). Since exp,(-qml(q- l))=O form= 1,2,3,..., then

exp((z- 1)log n)exp,( -x)d,x

(3.28)

for n>O, so that if 5 is a zero of exp,( -z), then

f,(z) = s 5 exp((z- 1)log x)exp,( -x)a,x. (3.29)

0

The dependence of f,(z) on log( l-q-‘) is relatively unimportant since

l?,(z) H exp( -2kz)f,(z) under 1-4-l H (1 -q-‘)exp(2ri). (3.30)

From Eq. (2.5)


On: Tue, 18 Oct 2016 06:17:31


exp, 5 =fi (l--q”-“)= mIJrl (1-q”) ( i n=l n=-co

(3.31)

so that

0

f-,(z)= c (4-l) *=-co

= c (q- 1) (; -p)z n=m+l n (1-C) m=O

q-1 g (q-z)” -ii (l-q-“). = qZ( 1 -cP)’ m=O n=mfl

(3.32)

Use of 4 integration by parts [or by direct substitution of 1 - ( 1 - q-*) for the extra factor 4 -m in any given term in Eq. (3.32)], gives

I=,tz+ 1)=4-‘(z)q-1~q(z)= -(-z),f,(z>. (3.33)

If z+(2Mri/log q) is substituted for z in Eq. (3.32), then

pq( z+ E) =fyZ)q-2M~Ww 1- ( q-1)-2Mdllogq=fq(Z)(~-q-l)-2M7rillogq

(3.34)

if MeZ. Let

II A ~q~~~=-~q~z~iO~~i-q-~~-qz~l~q-l~zm~o

q-1 g( mf l)tq-z)m log q fi (1 -q-“)

n=m+l

(3.35)

then the series for 6!(z) co_nverges absolutely for z such that R(z log q)>O. Since 6,(z) is the term-by-term derivatlvc of T,(z) with respect to z, and the series for $$(z) converges absolutely if !.R(z log q)>O, then r,(z) is differen$able for R(z log q)>O, and a,(z) is its derivative.

The functional equation (3.33) for I?,(. ) can be used to extend the function analytically to the entire complex plane. Since ~-‘(z)~-I = - ( - z), has simple zeros at 2MniAog 4, M EZ, then I’,(z) is a meromorphic function with simple poles at

2Mri z= -Nf -

1% 4 ’ M,NEZ, N==O.

Since

Fq(l)= lim I

qNNs- 1) M,N+m q

-M,(q-l) expq(-n)dq~=M!~~_[-expq(-x)l~~~~~"l,


On: Tue, 18 Oct 2016 06:17:31


=M!hyjexpq( $J -expq(s)) = 1 (3.37)

then

tq( 1+?!$j+q-*)-2Mnihgq. (3.38)

The residue of I?Jz) at z=-N+(2Mdlogq) is

,. @es r,(z)),= -N+(2M,,illog q)=

(-1)N(q-1)(1-q-1)-2M~illogq

WI,! log q . (3.39)

The proof is by induction on N, using Eq. (3.33). From Eq. (3.32), and Eq. (A7) in I, substituting q-* for q and q2 for z, then

a q-1 E (q-z)m fi (I-~-“)

b(z)= qz(l-q-‘)z m=O

n=m+1

‘41-y 1 -q-l)‘-z ij’ u-cYjc .;=yyq-“)

I-I;=,tl-q-“) =ql-zthP)l-z l-I;=o(l-q-(z+“)

1

tq-‘;q-‘)m =ql-‘( 1 -q-y (q-z.q-1)

3 m (3.40)

and so f,(z) has no zeros. If O<]q]<l, let

m lwqn+’ (4xL ~q(z)=q’-Zfq-,(z)=(l-q)‘-ZJ-J -= ___ n=O l-q”+z (1 -4)1-z (qz;q)m (3.41)

then

rqtz+ l)=tz),rqt2), r,(l)= 1 (3.42)

so that IT’,(z) can be regarded as a q-gamma function. Note also that

=rq(z)( 1 -q)-2M+m3 4. (3.43)

Note that Eq. (3.41) is the usual q-ga_mma function for O< 1qj-C 1 [Eq. (2.13) in Ref. 71. If ]q]> 1, let rqcz) = q(1/2)z(z- 1)rq(z), then

rqtz+ i)=(z)qrq(z), r,(i)= i (3.44)

so that T,(z) can be regarded as a q-gamma function. In this case


On: Tue, 18 Oct 2016 06:17:31


rq(z+?!$+)=(-l)M enp(2M7iir-~)rq~r)(l_q’)‘Minilogq. (3.45)

IV. g DEFORMATIONS OF THE GAMiblA FUNCTION: SYMMETRIC CASE

For O<lql<l, J# 0, the symmetric q integral

I -2N qz;c)= lim ’ ’

M,N+m q2M5 exp((z- 1)log x)E,( -x)d(x:q)

= lim g (4-‘-q)42m-’ M,N--’

5q~2m-l~~z-1~52-1~q(~q2m-l~)

m= -N+ 1

diverges as N -P ~4 if 5 $ q’/(l -q2> by Eq. (2.15). For specified choices of log q and lo& 1 - q2> [e.g., the principal value of log( 1 - q2>], the gamma function can be deformed to symmetric q-gamma functions (dependent on n) by setting

= lim M,N-+m ,,,= -N+ ,

= 5 (q-‘-q) 4’1”;;;);’ Eq( -;;-yj,

m=-co (4.2)

which converges if and only if R(z log q)CO, i.e., lqzl -C 1. [Note that integral converges for all z as N -P ~0, but as M + m if and only if R(z log q)<O.]

If n is even (= 2p, say), then

2P

i i

co

r; z+-+ = C (4-1-d

q(2m+2p-l)z

m=-co (1-q2)z Eq( -q;:-yq

= 2 (q-‘-q) y:““,:;; Eq( +$p), m=-cc

which is independent of p. Let I’:(z) = lT~(z;q2pl( 1 - q2)) (superscript e denoting “even”). Similarly, if n is odd (=2p+ 1, say), then

2p+1 Co

r; z&j-T = c W-4) i i

q(2m+2p)z -q2m+2p

m=-co (l-42)’ Eq ( 2) l-q

= i (q-'-d ,I",:,zEq($2), m=-co

(4.3)

(4.4)


On: Tue, 18 Oct 2016 06:17:31


which is independent of p. Let T:(z) = lY~(z;q2P+ ‘/( 1 - q2)) (superscript o denoting “odd”). The dependence of T:(z), I’:(z) on log( 1 - q2) are relatively unimportant since

I’:(z) H exp( -2niz)r;(z) under 1 -q2 H (1 -q2)exp(2Ti), (4.5)

r:(z) H exp( -2rCz)I’;(z) under 1 -q2 b-9 (1 -q2)exp(2k).

Note that r;(i)=r;(i)=i. Use of symmetric q integration by parts [or direct substitution for E,, using its symmetric

q-differential equation, into the infinite series (4.3) and (4.4)], gives

r;(z+ i)=[z],r;(z), rp+ i)=[z]qr;w (4.6)

and so

rp+ i)=rp+ i)=[ij,!, n d, do. (4.7)

By direct substitution of z + (Mdlog q) for z into the series for f’;(z) and T:(z), then

r;(z) ( 1 - qy~iw3 4 (4.8)

if M EZ, so that

t- ~~“C~lq! (1-q2)M~illogq~

blq! ( 1 -q2)Mrillog 4 (4.9)

for M, n EZ, n20. Let

cn

.n~(z)=-r~(z)i0g(i-q2)+ 2 (q-l-4) t2m- l)q (2m-‘)z log q

m=-co (l-42)” Eq

-q2m-1

i 1 l-q2 ’

(4.10) 2mq 2mz log q

~~tz)=-r~tz)l0gti-q2)+ 5 tq-l-q) t1-q2jz -qzm

m=-co Eq 1-42 ( i

then the series for R:(z) and O:(z) converge absolutely for z such that ‘X(z log q)<O. Since O:(z) and s28(z) are, respectively, the term-by-term derivatives of T;(z) and r:(z) with respect to z, and the series for n;(z) and a:(z) converge absolutely if %(z log q)<O, then T;(z) and r:(z) are differentiable for R(z log q)<O, and n:(z) and Q:(z) are their respective derivatives.

The functional equations (4.6) for r;( .) and I?:( .) can be used to extend the functions analytically to the entire complex plane. Since [z], has simple zeros at Mdlog q, M E .& then T;(z) and l?:(z) are meromorphic functions with simple poles at

Mri z= -N+ -

1% 4 ’ M,NEZ, NaO.

The residues of T:(z) and r:(z) at z= -N+(Mdlogq) are

t-11 M+N(q-q-‘) (Res rG(Z))z=-N+(Mniltog q)=2[N],! log q( 1 -q2)bfdbg q 7

(4.11)

(4.12)


On: Tue, 18 Oct 2016 06:17:31


The proof is by induction on N, using Eq. (4.6). Because of the behavior of r:(z) and l?:(z) at their common simple poles, the singularities

may be removed from some of the poles by defining

r,‘(z) = grgcd q(z)) for z EC, then lYt( z) are meromorphic functions, J?:(z) has simple poles at

(4.13)

2Mri z= -N+ -

1% 4 ’ M,NE Z, NaO,

with residues

and T,(z) has simple poles at

z=-N+(~;~;~~~, M,NEZ, N>O,

with residues

(4.14)

(4.15)

(4.16)

(- l)Nfl(q-q-l) (4.17)

Also, from Eq. (4.9), then

r; 2Mri

N+l+- = [Nl,!

1% 9 4 1 ( 1 _ q2)2Mvikz q

(4.18)

r, N+1+(2M+l)~i = (

-WI,! 1% 9 1 ( 1 _ q2)(2M+ l)lri/log q ’

for M,NEZ, NSO, so that r;(z) has zeros at N+(MrriAog q) for M,NeZ, M odd, N>O, and r,(z) has zeros at N+(Mrrillog q) for M,NEZ, M even, N>O.

From Eq. (4.6), then

(4.19)

and from Eq. (4.8), then

even,

odd. (4.20)

I ( 1 - q2)‘~?rillog 4 ’


On: Tue, 18 Oct 2016 06:17:31


Let F,(z)=T,+(z)T,+( 1 -z), then

F,(z+ l)= -F,(z), F.(z+ $g =F,(z) (4.2 1)

so that F,(z) is doubly periodic with periods 2 and 2 ai/log q. In a fundamental parallelogram, F,(z) has two simple poles, so that it is an elliptic function of order 2. There are two known noncongruent zeros [at 2N+((2M+ l)rri/log q) for some M,NEZ, and (2N+ 1)+((2M+ 1)rril log q) for some M,N EZ], so these are the only zeros, and they are simple zeros, i.e., T;(z) is a meromorphic function with its poles being simple poles at -N+ (Mmillog q) for M,N EZ, M even, and NaO, and its zeros being simple zeros at N+ (Mrriflog q) for M,N EZ, M odd, and N>O.

Let Gq(z)=r;(z)rq(l -z), then

G,(z+ l)= -G,(z), Gq( z+ s) =G,(z) (4.22)

so that G,(z) is doubly periodic with periods 2 and 27dlog q. In a fundamental parallelogram, G,(z) has two simple poles, so that it is an elliptic function of order 2. There are two known noncongruent zeros [at 2N+(2Mrriilogq) for some M,NEZ, and (2N+ 1)+(2Mdlogq) for some M,N EZ], so these are the only zeros, and they are simple zeros, i.e., T,(z) is a meromorphic function with its poles being simple poles at -N+ (Mdlog q) for M,N E Z, M odd, and N>O, and its zeros being simple zeros at N+ (Mrdlog q) for M,N EZ, M even, and N>O.

Let H,(z)=T,+(z)T;(l -z), then

H,(z+ l)=H,(z), .,(z+ z) =H,tz) (4.23)

so that H,(z) is doubly periodic with periods 1 and 2riAog q. The simple poles of r:(z) all coincide with zeros of I’,( 1 - z), and the simple poles of ri( 1 - z) all coincide with zeros of I’:(z), so that H,(z) has no singularities, and is therefore constant. Since neither I’;(z) nor T,(z) is identically zero, then H,= H,(z) is a nonzero constant. [It immediately follows that the zeros of T:(z) are simple zeros at z = N+ (Mm’llog q), for M,N EZ, M odd, and N>O, and that the zeros of T,(z) are simple zeros at z=N+(Mdlogq), for M,NEZ, Meven, and N>O.]

From Eq. (2.13), product expansions for the symmetric q-gamma functions, I?;(z), r;(z), T:(z), and T,(z), can be found. The product expansions are given by

(1-q2n+2)(l+q4n+3-2z)(1+q4n+1+2z)

(1-q 2(n+z))( 1 +q4n+l)( 1 +q4n+3)

(1 mq2n+2)( 1 +q4n+3-2z)( 1 +q4n+1+2z)

(1-q vn+z))( 1 +q2n+l) ’

(1 wq2n+2)(1 +q4n+3+2z)(l +q4n+l-2z)

(1 -q2(“+2))( 1 +qhn+‘)( 1 +q4n+y

(1-q2n+2)(1+q4n+3+2z)(l+q4n+‘-2z)

(1 -q2(“+z9( 1 +q2n+l) ’

(4.24)


On: Tue, 18 Oct 2016 06:17:31


1 qz-’ m c-(z)=z (l-q2)‘-’ n=O nl

(1 -q”fl)( 1 +qn+l-z)

I (1-q”fz)(l+q”+‘) ’

1 qz-1 m (,-,n+l)(1-,n+1-z) Vz)= z ( l-q2)z-l n=O ni (l+q”+z)(l+q”+t) )

(these results are proven in Appendix B), so that T:(z) has zeros at z=2N- f+ [(M+( 1/2))7ri/ logq], forM,NEZ, I’:(z) has zeros atz=2N+~+[(M+(1/2))~illogq], forM,NEZ, rqf(z) has zeros at z=N+[(2M+l)dlogq], for M,NEZ, N>O, and I?,(z) has zeros at z=N +(2Mdlogq), for M,NEZ, N>O.

It can reasonably be argued that as q + 1, lqla, agq + 0, r;(z), r;(z), r,+(z), all approach r(z), and I‘,(z) approaches 0. To see this, first note that it is reasonable that as q -+ 1, Iql-a wq -+ 0

m 1+aqn+b n n=O 1 +e?“+c -+ (1 +a)‘-*. (4.25)

One argument for Eq. (4.25) is that if Ial < 1, the logarithm of the relevant product is given by

m (- l)m+luyqbxqcm) c

m=l m(l-q”) ’ (4.26)

which has the series for (c-b)log( 1 +a) as its termwise limit. Another argument is that if b and c differ by an integer, then, by cancellation in the infinite product, ( 1 +a)c-b is the limit as 9 + 1~1q1~1~wzq --t 0, for general a. By the argument in Appendix A of Ref. 7, then

1 -qn+l

1-q”+z + l-(z) (4.27)

as q --+ 1, [q I < 1, and so it immediately follows from the first expressions for I’:(z) and r:(z) in Eq. (4.24) that they both approach T(z). The limit of r:(z) is given by

(4.28)

since the expression in the square bracket has limit 2 (the first factor has limit 2’-‘, and the product has limit 2’). The limit of T,(z) follows from the fact that every factor in the product has limit 0.

From Eqs. (4.21), (4.22), and (4.23), F,(z), G,(z), and H, are given by

m (l-q”+*)2(l+q”+‘-~)(l+q”f~) qz1= f w-d-I

1 I n=O (1-,~+~)(1-,~+1-~)(l+q~+Y ’

(,-qn+1)2(1-qn+l-z)(l-qn+z)

(1 +q”fz)( 1 +qn+l-r)( 1 +qn+y ’ I (4.29)


On: Tue, 18 Oct 2016 06:17:31


H,=$ (q-l-q)fi n=O [ i:~~~~~i:}=a(q-l-q)~, [s)2

=; (q-l-q)fi [( 1 -q2”)2( 1 -q2”-l)4] n=l

=&l-q) l

1+25 (-l)nqnZ ) 1

2

n=l

where the third equality for H, follows from Eq. (AlO) in I. Note that the function q - H, is an analytic function on the unit disc, with a simple pole at q = 0, no zeros, and a natural boundary at the unit circle.

Values of the symmetric q-gamma functions at other lattice points are

4 -1 -4

4[N],!( 1 -q2)2MTi”ogq

rl -N+ (2M+1)Ti 4-P 1% 4 =4[N]q!(1-q2)(2Mcl)~i’logq

(4.30)

for M,NEZ, NaO.

V. CONCLUSION

The properties and asymptotic behavior of the standard and symmetric q-exponential functions have been used to define families of q-gamma functions, and product decompositions for the q-gamma functions have been found. A one-element family in one case [see Eq. (3.41)] consists of the standard q-gamma function [Eq. (2.13) in Ref. 71. On the other hand, other q-gamma functions have been defined. These functions have not had much consideration previously. The q-gamma functions considered in this article are related to the generalized exponential function exp(z;q,X) defined in I, at the values, 0, 1, 1, for X. The existence of generalized gamma functions for other values of A remains an open problem.

One possible future development is to define and study a q-zeta function, or a family of such functions, as a q deformation of the Riemann zeta function for general q in the standard and symmetric cases (and possibly even more general X), and to relate these functions to the corre- sponding q-gamma functions in the same way as in the classical case.

ACKNOWLEDGMENTS

The author acknowledges the financial support of an Australian Postdoctoral Research Fel- lowship. He thanks A. J. Bracken and P. D. Jarvis for many interesting conversations and helpful suggestions.


On: Tue, 18 Oct 2016 06:17:31

D. S. McAnally: q-gamma functions

APPENDIX A: A PROOF OF EQUATION (3.18)

Equation (C 1) in Appendix C on page 115 of Ref. 7, is stated in that text as follows:

5 (~xz)J” (blu;q),(ut;q),(q/ut;q),(q;q), -= n=-m (bx), (q/u;q)m(blur;q),(b;q),(t;q), ’ Iw444 C-41)

so that, putting a = - u, and b = 0, then

i ( _ u;q)ntn= (- wqM -dwqMq;qL

n=-z (-q,u;q),(t;q), ’ @+l<L @+l<l9 u # 0

and so

,& (-uiG4 =

(-~t;q)m(-q~~~;q)m(q;q)m

3 m (-u;q)~(-qlu;q),(f;q), (A3)

if O<lql<l, O<lrl<l, u $ -qz, II f 0. Putting t=qZ and u=(l-q)[, then Eq. (3.18) follows, noting that [ $ - qz/( 1 -q), so that K $ - 4’.

Alternatively, let

Wt,q,u) = n (1 +utq”)( 1 +u-‘Pq”f’) 1 G(f,q,u) = i tm I-tq” 9

n=O m=-oo K=,U+qR4

(A4)

forO<lql<l, O<ltl<l, u # 0, u $ -4’. Since

and so

G(@,q,u)u= 5 qmtmu

m=-m qLA1 fq”u)

cc G(f,q,u) + G(qt,q,u)u = c

t”( 1 fq%)

m=-m c=,c 1 f-q”u)

tm

=,g Ky=,+,(l +q”u)

=,.-,. =m- ‘;;;qnu, =t-lG(fw) n--m

l-t t G(t,q,u)=uG(qt,t,u).

(‘46)

(A7)

Now


On: Tue, 18 Oct 2016 06:17:31


l-t t F(t,q,u)=

(l+ut).rI;=l(l+utq”).nl,m=O(l+u-~t-~q”+l)

d-I;= 1( 1 - tqy

(1 +u-lt-l).II;=o( 1 +utq”+‘).rII,“=,( 1 +u-Wq”) = l4

rI,“,,( 1 -tq”fl)

- 4 (1 futq”fl)( 1 +u-Vq”) =U

(l-tq”“) 1 =Nqt,q,u). n=l

648)

Since F( t,q,u) is analytic in the punctured unit disc (0~ I tl < 1 ), then the Laurent series expansion of F(t,q,u) for O<]tl< 1 is given by

Ftt,q,u)=dq,u)G(t,q,u)=g(q,u) 5 tm

m=-m l-c,t 1 fq”u) (A9)

Letting h(q,u)= llg(q,u), then

,$ nI”- (fLltqnu) =h(q,u)fi [ (l+utq’)(ll~+$-‘4”1’)] n--m n=O

(A101

for O<lql<l, O<ltl<l, u # 0, u $ -4’. {Alternatively, let

h(t,q,u) = Gtt,q,u) F(t,q,u) ’ O<]tl<l

then Wqt,q,u)=h(t,q,u) for O<Iql<l, O<]tl<l, u#O, u+ -42. Let hl(u,q,u)=h(exp(u),q,u), then h, is doubly periodic with periods 2rri and logq. In a fundamental parallelogram for h, , h, has at most one simple pole [at (2Mf l)~i+ N log q-log u, some M,N E Z], but there is no doubly periodic function with one simple pole in its fundamental parallelogram, so that hl (and hence h also) is constant.}

Now

N-l

Gtt,q,u) rl[ (1 -q”t)= 5 tm

m=--oo K=,(l+qnu).IT~~~-N(l+q-nu-l)’ NaO. 6412)

n=O

This can be proven by induction on N. If N= 0, then Eq. (A12) is true by the definition of G(t,q,u). If N> 1, then

N-l

G(t,q,u) T]I (1 -q”t)=( 1 -qN-5). i tm

n=l m=-cL, IT,“,,(l+q”u).~~~~-,+,(l+q-“u-l)

tm+q -m+NtmU- 1

m=-m KLdl +q”u).rI~~‘-N( 1 +q-“u-l)

-,-, I-I”- (1 +q34)qIT;:;+l(l +q-“U-l) n-m

J. Math. Phys., Vol. 38, No. 1, January 1995

Reuse of AIP Publishing content is subject to the terms: https://publishing.aip.org/authors/rights-and-permissions. Downloaded to IP: 130.102.82.2

On: Tue, 18 Oct 2016 06:17:31


=,-. l-I”- (l+~~~~~~~~(r:,n.i, n--m

-i 4

N-ltm

m=-m l-I&,-,(1 +qnu).rI;&Jl +q-“u-l)

=jim ?T- (l+q”u).rI;;eN(l +q-“u-l) * n-m (Al3)

This completes the proof by induction. Since

N-l

G(t,q,u) n (1 -q”t)= ;r, t”

,n=-m Kxl +qnU).n;~‘-,(l +q-“u-l) n=O (A14)

then

G(t,q,ki (1-q”t)= 5 tm

m=-m rII,“=,( 1 +qnu).rI~2,( 1 +q-“u-l)f G-W n=O

From Eq. (AIO), then

W.wii (1 -qQ)= I: tm n=O m=-m r-q=,(l +qnu).n;2,(1 +q-“u-l)

m

=h(q,u)n: [( 1 futq”)( 1 +.-lt-lqn+‘>] n=O

we

and so

G(r,qdi CU-qWl-q"+')l= i tmrI;,o( 1 -q”+‘)

n=O m=-m l-I;=,< 1 +qnu).rI~2,( 1 +q-“u-l)

cc

=h(q,u)fl [(1-q”f’)(l+utq”)(l+u-‘t-‘q”f’)l n=O

m

=h(q,u) 2 q(l/2)m(m-1)Umtm~ (A17)

Equating coefficients of tm, then


On: Tue, 18 Oct 2016 06:17:31


h(q,u)q(l/2)m(m-l)um= rr,“=,(l-d?

II,“=,< 1 +q”u).rI;z!,( 1 +q-“i’)

and so

9 m-‘urI~=l( 1 -q”> = rI,“=,-,( 1 +q”u>.n;~2,( 1 +q-“u-‘)

q(l~2hh-~)u~~~=l(1 -qn)

= rI;=,( 1 +q%).rI;:-,(l +q-“u-‘) (Al@

h(q,u) = Jx=,(l-47

II;=,( 1 +qnu).n;:-,( 1 +q-“u-l) =

(z,“,-,( _ 1 )P~UW&J~+P))~

z;=-, q(1’2)m(m-1)um * 6419)

Therefor<

for O<lql,

,*, nm- ;,+qnu) = ii [(l ~lu~~~!t(~~~ir:4::;:jtSqr;il)] n-m n=O

<l,O<ltl<l,u # 0,u $ -92.

G420)

Setting t=q‘, and u=(l -q)L, then we get Eq. (3.18).

APPENDIX B: PRODUCT FORMULAS FOR *GAMMA FUNCTIONS

If O<A<l, B>O, and O<C< 1, then the double sum ZF=;=_, XI=-k Ak2BkCn converges. This is seen as follows.

k=-oo ,,=-k k=-co 1-c

l-I;=,[( 1 -A2”)( 1 +A2”-‘BC-‘)( 1 +A2”-‘B-‘C)] =

1-c Cm. (Bl)

If R(z log q)<O, then from Eq. (2.13)

r;(z)= r, (q-l-q) ;;““;;;z E, n=-cc

,,= -co k=O

4 “-‘c;=-, C~=‘=_nq2nzqk(2k+l)n;k+n+l( I-421) =

(1 -qyc;=-, qzm2+, .

The summand in the double sum in the numerator satisfies

W)


On: Tue, 18 Oct 2016 06:17:31


co m

42nzQk(2k+ l) n: (l-421) <lqzplqp+l) rl] (1 +lq12’) I=k+nf 1 l=k+nfl

4i (1 +lq12’M2kZlqlklqz12n l=l

(B3)

and so the double sum converges absolutely as 0 -=c I q I 2 < 1, 1 q I > 0, and 0 -=c Iq’/ 2 < 1. Therefore

r;(z) = 4 ‘-‘z;-, gLk q2nzqk(2k+l)~I;3_k+n+l(l -42’)

(1 -qy*z;,-, q2m2+m

4 z-*c;=‘=_, q. q2nz-2kzqk(2k+1)~I;O=n+l( 1 -q2L) =

(1 -qyc;,-, q2m2+m

4 z-‘~;km=-m q-2kzqk(2k+‘)C;=o q2nTI;a_n+l( l-421) =

(1 -qyz;=-, qzm2+,

4 Z-‘n;==,(l -q2y~;z’=_, q-2kZqU2k+‘) m

= (1 -qy2;=-, qzm2+, ‘Z. rI;=,;T42’)

qz-‘rI;~‘(l -q2[).C;=+ q-2kzqk(2k+‘)

= (1 -q2)+I;=o( 1 -q2n+2z).x;=-m qZm2+,’

where the last equality follows from Eq. (A7) in I, substituting qz for z, and q2 for q. By Eq. (A8) in I (Jacobi’s triple product formula), then

1 r;(z) 4 Z3-I;“=,( -q2’).c;=-, q-2kzqk(2k+‘) =

(1 -qyn;=o( 1 -q2n+2z).x;=-m qzm2+,

2-l Co (1-q

= uTq2)z-cn=o l-I

2(n+1))(~-~4n+4)(1+~4n+3-2z)(l+~4~+l+2~)

(1 -qz(“+z))( 1 -qdn+4)( 1 +qdn+‘)( 1 +q4”+3)

z-1 -

= (1 Tq2Y-f n=O I-I

(1 -eq2n+2)(1 +q4n+3-2z)(l +q4n+1+2z)

(1 -qz(“+z))( 1 +q4”+‘)(1 +q4n+3)

yielding the product formula for T:(z) in Eq. (4.24). The product formula for l?:(z) is proven similarly. If R(z log q)<O, then

= i (q-‘-q) ,$;;:;z K,,( ;<nq:‘)k I: (q-‘-q) ,l”;lz 44 &i) p$= --2 n=-cc

= i (k l)‘(q-l-q) (;Tq;;z El(s) a= --m


On: Tue, 18 Oct 2016 06:17:31


= (;;y;qjg’ 5 (+ l)nqnz x;co 4i”2’(~~~~~-2~~~~=k+“1 -q2’)

2 2 n=-cc m- cc

4 Z-l~;=--m ~:kp=~( + l)“q”zq(‘/2)(n-2k)(“-2L-l)n;o=k+l(l -q21) =

(1 -qy2;=-, q2m2+m

4 (1/2)(.-k)(n-r--‘)n~=~“~~(l -q[) =

(1 -q2)z-‘c;,-, q2m2+m

4 (“2)k(k+‘)n;“=:~“~n,2( 1 -ql) =

(1 -qy2;=-, q2m2+m 036)

The summand in the double sum in the numerator satisfies

m co qnzq(l/2)k(k+ 1) IT (l-qg <Iqzjnlql(“2)k(k+‘) JJ (l+lql’)

l=k+n+2 l=k+n+2 I even 1 even

s J-J (1 + lql~).~q~(“2)‘2~ql(“2)k1421” 1=2

037)

I even

and so the double sum converges absolutely as 0 < 141 “2< 1, and 0 < lqzl < 1. Therefore

4 2r;(z)=

(“2)k(k+ ‘)&lk;v;n+2( 1 - ql)

(1 -qyc;,-, qQ+m

4 n=-k k+n=O (mod 2)

(” ~)kq”zq(“2~k(k+‘~nP=~~~~2( 1-q’) =

(1 -q2)z-‘2;=-, q2m2+m

4 ‘-‘c;==_, z”n=o (+ l)kq”zq-‘zq(“2)k(‘+‘)~~=~“~~( 1 -q[) n even =

(1 -qyc;,-, q2m2+m

4 ‘-‘z;==_, C,“=,(k l)kq2”Zq-kzq(“2)k(k+‘)~~=“+,( l-$[) = . .._ -

(1 -qy~;=-, q2m2+m

4 z-‘n;“=,( 1 -q2’)~;z’=_,(t l)kq(‘/2)k2+(‘/2)k-kz

= (1 -q2)91,“,o( 1 -q2n+2z)x;=-m q2m2+m ’ 038)

where the last equality follows from Eq. (A7) in I, substituting qz for z and q2 for q. By Eq. (A8) in I (Jacobi’s triple product formula), then


On: Tue, 18 Oct 2016 06:17:31


2r;(z) = ’ z-‘n[;~,~(l -q21)~;E’=_,(t l)kq(“2)k2+(‘/2)k-kz

(1 -q2)+l;=o( 1 -q2n+2z)z;=-m qzm2+,

z-l

= (1 ‘q2)z-’ n=O Ii (1-q an+‘))( 1 -qn+‘)( 1 +qn+‘-z)( 1 t,,+z>

(1 -qZ(n+z))( 1 -qdn+4)( 1 +qbn+‘)( 1 +qdn+3)

z-l

ii (1 -q”+‘)( 1 tqn+‘-Z)

= (1442)z-r~Eo (l+q”+~)(l+q2”+2)(1+q4~+‘)(1+q4~+~)

z-l fi (1 -q”+‘)( 1 tq”+l-‘) = (l&l n=O (lTq”+Z)(l+qn+‘) ’ @9)

where the final equality follows from the fact that each positive integer must be even, or 1 or 3 more than a multiple of 4. The product formulas (4.24) for I?,‘(z) follow automatically.

’ D. S. McAnally, J. Math. Phys. 36, 546 (1995). *L. C. Biedenharn, J. Phys. A 22, L873 (1989). ‘A. J. Macfarlane, J. Phys. A 22, 4581 (1989). ‘A. J. Bracken, D. S. McAnally, M. D. Gould, and R. B. Zhang, J. Phys. A 24, 1379 (1991). ‘D. Bernard and A. Le Clair, Phys. Len. B 227,417 (1989). 6P. P. Kulish and E. V. Damaskinsky, J. Phys. A 23, L415 (1990). ‘G. E. Andrews, q-Series: Their Development and Application in Analysis, Number Theory Combinatorics, Physics, and

Computer Algebra, published for the Conference Board of the Mathematical Sciences (The American Mathematical Society, Providence, RI, 1986).


On: Tue, 18 Oct 2016 06:17:31

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