Aq-Raabe o mula and an in eg al o he
ou h Jacobi he a unc ion
Is ´an Mez˝o
Escuela Poli ´ecnica Nacional,
Depa amen o de Ma em´a ica,
Lad ´on de Gue a a E11-253, Qui o, Ecuado
Abs ac
We gene alize he Raabe- o mula o he q-loggamma unc ion by gi ing an in eg al
o mula o log Γqwhen q > 1. As a consequence, we ge ha he in eg al o he
loga i hm o he ou h Jacobi he a unc ion be ween i s leas imagina y ze os is
connec ed o he pa i ion unc ion and he Riemann ze a unc ion.
Key wo ds: q-gamma unc ion, q-loggamma unc ion, Jacobi he a unc ions,
hype geome ic unc ion, Riemann ze a unc ion, pa i ion unc ion,
Raabe- o mula
1991 MSC: 33E05, 33D05
1 In oduc ion
The ou h Jacobi he a unc ion is de ined by he in ini e sums
ϑ4(x, q) = ∞
X
n=−∞
(−1)nqn2e2nix = 1 + 2 ∞
X
n=1
(−1)nqn2cos(2nx).
This is an en i e unc ion in he complex a iable x o any ixed complex q
o which |q|<1. See [4,25] on he he a unc ions in gene al.
The pa i ion unc ion P(n) gi es he numbe o possible addi i e in ege
pa i ions o he na u al numbe n. In o he wo ds, P(n) is he numbe o
ways w i ing nas a sum o posi i e in ege s [3,11].
Email add ess: [email p o ec ed] (Is ´an Mez˝o).
P ep in submi ed o Jou nal o Numbe Theo y 29 Augus 2012
In he p esen pape we e eal an analy ic connec ion be ween he ou h
Jacobi unc ion and he pa i ion unc ion.
I is easy o p o e ha o a ixed eal q∈]0,1[ he unc ion ϑ4(ix, q) is eal
and s ic ly posi i e when x∈] log √q, −log √q[, and i is ze o a hese wo
endpoin s (see he inal ema ks o Sec ion 4). Ou main esul conce ns on
he a ea unde he g aph o he ou h Jacobi unc ion on his in e al ( he
g aph can be ound in he penul ima e sec ion).
Theo em 1 Fo any eal q∈]0,1[,
Zx∗
−x∗
log ϑ4(ix, q)dx =ζ(2) −log q·log ∞
X
n=0
P(n)q2n!.
He e x∗= log √q,P(n)is he pa i ion unc ion, and i=√−1.
(No e ha x∗is nega i e and −x∗is posi i e, so in he heo em he in eg a ion
goes om ” igh o le ” on he imagina y axis.)
To p o e he heo em we need a q-analogue o Raabe’s in eg al
Z1
0log Γ(x)dx = log √2π
o he Eule gamma unc ion. This analogue uses he so-called q-gamma unc-
ion and s a es:
Z1
0log Γq(x)dx =ζ(2)
log q+ log
u
u
q−1
6
√q+ log(q−1;q−1)∞(q > 1),
whe e (q−1;q−1)∞=Q∞
n=0(1 −1/qn).
Fo a mo e gene al s a emen , see Theo em 2. The de ini ion o Γqis gi en in
equa ion (2).
2 P elimina ies
In his sec ion we se up all he necessa y ing edien s and p elimina y p opo-
si ions o p o e ou q-Raabe o mula and he in eg al o mula.
2
2.1 Raabe’s o mula
In 1840 J. L. Raabe [22] p o ed ha o he Eule Γ unc ion
Z1
0log Γ(x+ )dx = log √2π+ log − ( ≥0).
This implies he special case (when we ake he limi →0+)
Z1
0log Γ(x)dx = log √2π,
and an immedia e consequence is ha
Z1
0log Γ(x)Γ(1 −x)dx = log 2π.
(See [1] o an elemen a y p oo o his special case.) We shall p o e he app o-
p ia e in eg al o mula o he q-gamma unc ion when q > 1 (Theo em 2.).
Then we show ha he Jacobi iple p oduc iden i y connec s he q-gamma
unc ion o ϑ4and ou main o mula (Theo em 1.) will ollow.
To ead mo e on he Raabe- o mula and i s ex ension o mul idimensional
case, he eade may consul [16,23].
2.2 The q-gamma unc ions
F. H. Jackson de ined, o 0 < q < 1, he q-analogue o he s anda d Eule
Γ(x) unc ion o any x∈R {0,−1,−2, . . . }as [8,13,12]
Γq(x) = (q;q)∞
(qx;q)∞
(1 −q)1−x(0 < q < 1) (1)
wi h he so-called q-Pochhamme symbol (x;q)∞= (1−x)(1−qx)(1−q2x)···.
This Γq unc ion is called as Jackson q-gamma unc ion. This plays an impo -
an ole in he e alua ion o basic hype geome ic se ies [8]. R. Askey also
con ibu ed o he Jackson q-gamma unc ion in a p o ound way, see [5,6]
o examples. On analy ic p ope ies o Γq(including in o ma ion on poles,
esidues, in ini e sum ep esen a ions) one can u n o he book [24, Sec ion
6.4].
Ano he q-gamma unc ion can also be de ined o q > 1. I is
Γq(x) = (q−1;q−1)∞
(q−x;q−1)∞
(q−1)1−xq(x
2)(q > 1).(2)
3
This unc ion was in oduced by Jackson [12, p. 129], bu he did no s udy
i s p ope ies. The e a e wo undamen al pape s o D. S. Moak [20,21], in
which he in es iga ed i s analy ic p ope ies (see also Exe cise 1.23 o [8] and
Exe cise 14 on p. 546 in [24]). The e o e in conside ing his con ibu ion o
Moak, one migh call he unc ion in (2) he Moak q-gamma unc ion.5
We emphasize ha in he p esen pape we need and use only he q-gamma
unc ion when q > 1.
2.3 The ze a egula ized p oduc
We also need some ecen esul s o Ku okawa and Wakayama. Le us conside
a sequence a= (a1, a2, . . . ). I s ze a egula ized p oduc is deno ed and de ined
by [14,15]
[
∞
Y
n=1
an= exp −Res
s=0
ζa(s)
s2!.(3)
He e
ζa(s) = ∞
X
n=1
a−s
n
is he ze a unc ion associa ed wi h he sequence a. I is assumed ha ζa(s)
is me omo phic a s= 0 o a leas i can be me omo phically con inued o
s= 0, and u he , ha a ound his poin ζa(s) has he Lau en expansion
ζa(s) = X
m>m0
cm(a)sm
o some in ege m0. Thus he ze a egula ized p oduc equals o exp(−c1(a)),
as well.
See [19] o nice applica ions o egula ized p oduc s.
M. Le ch’s o mula (5) implies ha o a= (x, 1 + x, 2 + x, . . . )
[
∞
Y
n=0
(n+x) = √2π
Γ(x)(x > 0).(4)
The sequence aabo e has he associa ed ze a unc ion
ζa(s) = ζ(s, x) = ∞
X
n=0
(n+x)−s(x > 0,<(s)>1).
This is he well known Hu wi z ze a unc ion. Wha Le ch p o ed is ha [17]
ζ0(0, x) = log Γ(x)
√2π(x > 0).(5)
4
Then (4) easily ollows. See [2, Theo em 1.3.4] o a p oo o (5).
Now we s ep o wa d o he q- e sion o he abo e heo ems. Le us in oduce
he sho and s anda d no a ion
[n]q=qn−1
q−1(q6= 1).
Wi h his abb e ia ion he ze a unc ion associa ed wi h he sequence a=
([x],[1 + x]q,[2 + x]q, . . . ) is he so-called q-Hu wi z ze a unc ion:
ζa(s) = ζq(s, x) = ∞
X
n=0
[n+x]−s
q.
The e a e di e en q-ex ensions o he o dina y Hu wi z ze a unc ion, see [24]
o di e en examples.
Since he analy ical p ope ies and con e gence domains o his unc ion a e
c ucial in he p esen in es iga ion, we o mula e he nex p oposi ion.
P oposi ion 1 Fo any ixed q > 1 he q-Hu wi z ze a unc ion
ζq(s, x) = ∞
X
n=0
[n+x]−s
q
con e ges when x > 0and <(s)>0. A he poin s= 0 his unc ion has a
simple pole wi h esidue 1/log(q).
The p oo o his p oposi ion can be ound in he las sec ion.
Wi h espec o he q-gamma unc ion, he pa allel esul o (4) is in he nex
p oposi ion.
P oposi ion 2 Fo any eal q > 1and x > 0, he e holds
[
∞
Y
n=0
[n+x]q=
[
∞
Y
n=1
qn+x−1
q−1=Cq
Γq(x),(6)
whe e
Cq=q−1
12 (q−1)1
2−log(q−1)
2 log q(q−1;q−1)∞.(7)
This is he second heo em o Ku okawa and Wakayama in [14] and his will
be ou main ool. (A mo e gene al o m o his heo em is p esen ed in [19].)
Fo p ac ical easons we eph ase his ze a egula iza ion heo em in a mo e
sui able o m (employing (3) and (6)).
5

P oposi ion 3 Fo any q > 1and x > 0, he e holds
log Γq(x) = log Cq+ Res
s=0
ζq(s, x)
s2,
whe e Cqis de ined by (7).
The s a emen ollows om P oposi ion 2 by using he egula iza ion o mula
(3).
We spli he p oo o he main heo em o wo sec ions. The nex one con ains
he gene alized Raabe’s o mula, he o he con ains he p oo o he in eg al
o mula o Jacobi’s unc ion ϑ4.
3 In eg al o he q-loggamma unc ion – he q-Raabe o mula
The q-analogue o Raabe’s heo em o q > 1 is gi en:
Theo em 2 I q > 1and Γq(x)is de ined by (2), hen o any > 0,
Z1
0log Γq(x+ )dx = (8)
log Cq−1
2q log q"1−q
1−q− (2 Li2(q− ) + log2(1 −q− ))+
21−q
1−q− log 1−q
1−q log(1 −q− )−q log21−q
1−q #.
In pa icula , i ends o ze o hen
Z1
0log Γq(x)dx =ζ(2)
log q+ log
u
u
q−1
6
√q+ log(q−1;q−1)∞.(9)
He e Li2(z) is he diloga i hm unc ion [18]:
Li2(z) = ∞
X
n=1
zn
n2.
I is an in e es ing ques ion ha how such a heo em looks like when we use
he Jackson q-gamma unc ion (i.e., de ini ion (1) and 0 < q < 1). To look o
a heo em o his la ou , ou p oo canno be applied, since he wo c ucial
poin s – he q-Hu wi z ze a and he Ku okawa-Wakayama heo em – wo k
only when q > 1.
6
To p o e Theo em 2, we need he ollowing s a emen on he in eg al o he
q-Hu wi z ze a unc ion, which is in e es ing in i sel .
Theo em 3 I q > 1, > 0and <(s)>0, hen
Z1
0ζq(s, x + )dx =(q−1)s
slog q
(q −1)1−s
q 2F1(1,1; s+ 1; q− ).
He e
2F1(a, b;c;z) = ∞
X
n=0
(a)n(b)n
(c)n
zn
n!
is a hype geome ic unc ion and (a)n=a(a+1) ···(a+n−1) is he Pochham-
me symbol. See a good in oduc ion o hype geome ic unc ions in [10].
P oo o Theo em 3. To use he se ies ep esen a ion o he q-Hu wi z ze a
unc ion, we ha e o assume ha > 0, q > 1 and <(s)>0 (see P oposi ion
1). Then we ha e ha
Z1
0ζq(s, x + )dx =Z1
0
∞
X
n=0
[n+x+ ]−s
qdx =
(q−1)s∞
X
n=0 Z1
0(qn+x+ −1)−sdx.
This la e in eg al can be compu ed i we de e mine he Taylo se ies o he
in eg and wi h espec o he a iable xand hen we in eg a e e m by e m.
A leng hy compu a ion inally shows ha he in eg al can be exp essed by
hype geome ic unc ions:
Z1
0(qn+x+ −1)−sdx =1
slog q"(qn+ −1)1−s
qn+ 2F1(1,1; s+ 1; q−n− )−
(qn+ +1 −1)1−s
qn+ +1 2F1(1,1; s+ 1; q−n− −1)#.
(I can be ealized ha he abo e sub ac ion comes om he New on-Leibniz
o mula, so one can ead ou he p imi i e unc ion and hen check his in eg al
by de i a ion, oo.) Since
2F1(1,1; s+ 1; q−n− ) = ∞
X
k=0
k!
(s+ 1)k
1
(qn+ )k,
and because o <(s)>0,
∞
X
n=0 Z1
0(qn+x+ −1)−sdx =
7
1
slog q
∞
X
n=0
∞
X
k=0
k!
(s+ 1)k (qn+ −1)1−s
(qn+ )k+1 −(qn+ +1 −1)1−s
(qn+ +1)k+1 !.
I we in e change he o de o he summa ion – which can be done by absolu e
con e gence –, we see ha he sum o e nis elescopic, so he only one e m
which no cancels belongs o n= 0. Thus he abo e exp ession simpli ies o
1
slog q
∞
X
k=0
k!
(s+ 1)k
(q −1)1−s
(q )k+1 =(q −1)1−s
sq log q
∞
X
k=0
k!
(s+ 1)k
1
(q )k.
This la e sum is again hype geome ic wi h pa ame e s (1,1; s+1; q− ), hence
we ge ou Theo em. 2
P oo o Theo em 2. P oposi ion 3 gi es ha
Z1
0log Γq(x+ )dx = log Cq+Z1
0Res
s=0
ζ(s, x + )
s2dx.
Since he esidue is aken wi h espec o s, we can ca y ou i be o e he
in eg al. Hence, by Theo em 3,
Z1
0log Γq(x+ )dx = log Cq+ Res
s=0
(q−1)s
s3log q
(q −1)1−s
q 2F1(1,1; s+ 1; q− ).
The esidue can be calcula ed as ollows: we lea e s3in he denomina o ,
hen we look o he coe icien o s2in he Taylo expansion o he emaining
unc ion. A lengh y calcula ion shows ha he esidue equals o
−1
2q log q"(1 −q )∂2
∂s22F1(1,1; s;q− )s=1
+ (10)
2(1 −q ) log 1−q
1−q
∂
∂s2F1(1,1; s;q− )s=1 −q log21−q
1−q #.
Now we deal wi h he pa ial de i a i es. Symbolically,
∂n
∂sn2F1(1,1; s;z) = ∞
X
n=0
(1)n(1)n
∂n
∂sn
1
(s)n
zn
n!.(11)
The Pochhamme symbol can be ew i en wi h he Γ unc ion:
(s)n=Γ(s+n)
Γ(s),
whence ∂
∂s
1
(s)n
=−1
(s)n
(ψ(s+n)−ψ(s)),(12)
and ∂2
∂s2
1
(s)n
=(ψ(s+n)−ψ(s))2
(s)n−ψ0(s+n)−ψ0(s)
(s)n
.
8
He e
ψ(z) = Γ0(z)
Γ(z)
is he digamma unc ion [5,9]. When nis a posi i e in ege , hen [2, p. 13]
ψ(n) = 1
1+1
2+···+1
n−1−γ=Hn−1−γ, (13)
and
ψ0(n) = −1
12−1
22−···− 1
(n−1)2+ζ(2) = −Hn−1,2+ζ(2).
(Hnand Hn,2a e he ha monic- and second o de ha monic numbe s, espec-
i ely. H0=H0,2= 0.) Now (11), (12) and (13) gi es ha
∂
∂s2F1(1,1; s;z)s=1
=∞
X
n=0
n!n!−Hn
n!
zn
n!=−∞
X
n=0
Hnzn=log(1 −z)
1−z.
The las equali y is s aigh o wa d (see [10]). Simila ly, o he second o de
de i a i e
∂2
∂s22F1(1,1; s;z)s=1
=∞
X
n=0
n!n! H2
n
n!+Hn,2
n!!zn
n!=∞
X
n=1
H2
nzn+∞
X
n=1
Hn,2zn.
(14)
By Cauchy’s p oduc , he la e sum is simply
1
1−z
∞
X
n=1
zn
n2=Li2(z)
1−z.
The i s sum can be de e mined easily. No e ha
H2
n−1=Hn−1
n2
=H2
n+1
n2−2Hn
n,
whence ∞
X
n=1
H2
n−1zn=∞
X
n=1
H2
nzn+∞
X
n=1
zn
n2−2∞
X
n=1
Hn
nzn.(15)
The las sum equals o [7]
∞
X
n=1
Hn
nzn= Li2(z) + 1
2log2(1 −z).(16)
I we empo a ily in oduce he unc ion
(z) = ∞
X
n=1
H2
nzn,
9