Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers

0.27 MB·

by  Mark W. Coffey

Bernoulli identities, zeta relations, determinant
expressions, Mellin transforms, and representation
of the Hurwitz numbers
6
1
0 Mark W. Coffey
2
Department of Physics
n
a Colorado School of Mines
J
7 Golden, CO 80401

USA
]
T
mcoff[email protected]
N
.
h
January 2, 2016
t
a
m
[ Abstract

1 The Riemann zeta identity at even integers of Lettington, along with his
v
other Bernoulli and zeta relations, are generalized. Other corresponding recur-
3
7 rences and determinant relations are illustrated. Another consequence is the
6
application tosumsofdoublezetavalues. Asetofidentities fortheRamanujan
1
0 and generalized Ramanujan polynomials is presented. An alternative proof of
. Lettington’s identity is provided, together with its generalizations to the Hur-
1
0 witz and Lerch zeta functions, hence to Dirichlet L series, to Eisenstein series,
6 and to general Mellin transforms.
1
The Hurwitz numbers H˜ occur in the Laurent expansion about the origin
: n
v
ofacertainWeierstrass℘functionforasquarelattice, andarehighlyanalogous
i
X totheBernoullinumbers. Anintegral representation of theLaurentcoefficients
r about the origin for general ℘ functions, and for these numbers in particular,
a
is presented. As a Corollary, the asymptotic form of the Hurwitz numbers
is determined. In addition, a series representation of the Hurwitz numbers is
given, as well as a new recurrence.
Key words and phrases
Bernoulli number, Bernoulli polynomial, Riemann zeta function, Euler number,
Euler polynomial, alternating zeta function, double zeta values, Hurwitz zeta func-
tion, Lerch zeta function, polygamma function, Ramanujan polynomial, Bernoulli
relations, zeta identities, Eisenstein series, recurrence, Hessenberg determinant, inte-
gral representation, Mellin transform, Hurwitz numbers
2010 MSC numbers
11B68, 11C20, 11M06
1
Introduction and statement of results
Let ζ(s) denote the Riemann zeta function and B (x) the nth degree Bernoulli
n
polynomial, such that B = B (0) = ( 1)n−1nζ(1 n) is the nth Bernoulli number
n n
− −
(e.g., [7, 18]). This relation is extended, for instance when n is even, to B (x) =
n
nζ(1 n,x) where ζ(s,x) = ∞ (n+x)−s (Re s > 1) is the Hurwitz zeta function.
− − n=0
There is the well known relatiPon (explicit evaluation)
( 1)m+1
ζ(2m) = − 22m−1π2mB . (1.1)
2m
(2m)!
The Bernoulli numbers have the well known exponential generating function
x B
= nxn, x < 2π,
ex 1 n! | |
− n≥0
X
while the ordinary generating series β(x) = B xn = 1 x/2+x2/6 x4/30+
n≥0 n − −
x6/42 + ... is divergent. However the contiPnued fractions for the generating series
B (4x)n, β(x), (2n+1)B xn, and (4n 2n) B x2n−1/n are con-
n≥1 2n n≥1 2n n≥1 − | n|
Pvergent [12]. P P
In[19]waspresentedanewidentityfortheRiemannzetafunctionatevenintegers,
jπ2j j−1 ( 1)kπ2j−2k
ζ(2j) = ( 1)j+1 + − ζ(2k) . (1.2)
− (2j +1)! (2j 2k +1)!
" #
k=1 −
X
This is hardly an isolated identity, and we show how to systematically derive many
related ones.
As an illustration, we have the identities given in the following.
Theorem 1. (a)
( 1)jj(2π)2j j ( 1)m+1
(31−2j 1)ζ(2j) = − +( 1)j+1(2π)2j − ζ(2m),
− (2j)!32j−1 − (2j 2m)!22m−132j−2mπ2m
m=0 −
X
(b)
j
( 1)jj(2π)2j ( 1)m+1
(41−2j 21−2j)ζ(2j) = − +( 1)j+1(2π)2j − ζ(2m),
− (2j)!42j−1 − (2j 2m)!22m−142j−2mπ2m
m=0 −
X
2
(c)
j
( 1)jj(2π)2j ( 1)m+1
(61−2j 31−2j 21−2j+1)ζ(2j) = − +( 1)j+1(2π)2j − ζ(2m),
− − (2j)!62j−1 − (2j 2m)!22m−162j−2mπ2m
m=0 −
X
(d)
j
j (2j)! ( 1)m+1 ζ(2m)
(1+72j−1)+ − (1+72j−2m)
−122j−1 (2j 2m)!22m−1π2m 122j−2m
m=0 −
X
(61−2j 31−2j 21−2j +1)(2j)!( 1)j+1
= − − − ζ(2j),
24j−1π2j
and (e)
j
j (2j)! ( 1)m+1 ζ(2m) (41−2j 21−2j)(2j)!( 1)j+1
(1+52j−1)+ − (1+52j−2m) = − − ζ(2j).
−82j−1 (2j 2m)!22m−1π2m 82j−2m 24j−1π2j
m=0 −
X
Moreover, inaseparatesectionwegiveanalternative proofof(1.2), anditsextensions
to ζ(s,a) and so to Dirichlet L-functions.
Now define functions
ζ(s) 1
φ (s) = , θ (s) = 1 ζ(s),
j js 3 − 3s−1
(cid:18) (cid:19)
1 1 1 1 1
θ (s) = ζ(s), θ (s) = 1+ + ζ(s). (1.3)
4 2s−1 − 4s−1 6 − 2s−1 3s−1 − 6s−1
(cid:18) (cid:19) (cid:18) (cid:19)
Theorem 1 and similar results have numerous implications for these and other func-
tions. In particular, Theorem 1(a) leads to Theorem 2(b).
Theorem 2. (a)
j2s π2s(2s 1) s−1 ( 1)s−nπ2n j2(s−n)
4 φ (2s) = − + − 4 φ (2s 2n),
22s j (2s+1)! (2n+1)! 22(s−n) j −
n=1
X
(b)
22j (1 3j) j−1 ( 1)m−1 2π 2m 3 −1
θ (2j) = ( 1)jπ2j − +2 − 1 θ (2j 2m),
3 − 32j (2j)! (2m)! 3 − 32(j−m) 3 −
m=0 (cid:18) (cid:19) (cid:18) (cid:19)
X
(c)
1 (1 4j) j−1 ( 1)m−1 π 2m 2 4 −1
θ (2j) = ( 1)jπ2j − +2 − θ (2j 2m),
4 − 22j (2j)! (2m)! 2 22(j−m) − 42(j−m) 4 −
mX=0 (cid:16) (cid:17) (cid:18) (cid:19)
3
and (d)
1 (1 6j) j−1 ( 1)m−1 π 2m 2 3 6 −1
θ (2j) = ( 1)jπ2j − +2 − 1+ +
6 − 32j (2j)! (2m)! 3 − 22(j−m) 32(j−m) − 42(j−m)
mX=0 (cid:16) (cid:17) (cid:18) (cid:19)
θ (2j 2m).
6
× −
We next recall the definition of a half-weighted minor corner layered determinant
Ψ (~h,H~), using the vectors ~h = (h ,h ,h ,...) and H~ = (H ,H ,H ,...):
s 1 2 3 1 2 3
H 1 0 0 ... 0
1
H h 1 0 ... 0
(cid:12) 2 1 (cid:12)
(cid:12) H h h 1 ... 0 (cid:12)
Ψs(~h,H~) = (−1)s(cid:12)(cid:12)(cid:12) ...3 ...2 ...1 ... ... ... (cid:12)(cid:12)(cid:12).
(cid:12) (cid:12)
(cid:12) H h h h ... 1 (cid:12)
s−1 s−2 s−3 s−4
(cid:12) (cid:12)
(cid:12) H h h h ... h (cid:12)
s s−1 s−2 s−3 1
(cid:12) (cid:12)
(cid:12) (cid:12)
Theorem 3. (a) Define vec(cid:12)tors ~u and U~ with entries u = (cid:12)1/(2s + 1)! and
(cid:12) 2 s (cid:12)
U = (2s 1)/(2s+1)!. Then
2s
−
j2s
4 φ (2s) = ( 1)sπ2sΨ (~u,U~ ).
22s j − s 2
(b) Define vectors U~ and H~ with entries U = 1/(2s)! and
3 3 3s
−2s
1 3s 2π
H = − +θ (2s)( 1)s−1 .
3s 3
− (2s)! − 3
" #
(cid:18) (cid:19)
Then
2s −1
2π 3
( 1)s Ψ (U~ ,H~ ) = 2θ (2s) 1 .
− 3 s 3 3 3 − 32s
(cid:18) (cid:19) (cid:18) (cid:19)
Let t denote the multinomial coefficient, the coefficient of the expansion
d1,d2,...,ds
of the su(cid:0)m of s t(cid:1)erms to the tth power. 1 2
Corollary 1.
(2π)2s 1 s t ( 1)t+s
4φ (2s) = − ,
j
j2s (22s−1 −1) Xt=1 dXi≥0(cid:18)d1,d2,...,ds(cid:19)3!d15!d2 ···(2s+1)!ds
1On pp. 22 and 23 of [20], the following typographical errors occur. The upper index for the
multinomialcoefficientforthe summations forη(2s), φ(2s), andθ(2s)inLemma 3.3shouldbe t. In
the display equation for the proof of Theorem 1.3, an “=” should be inserted after x2s−2.
2Onp.17of[20],( 1)k shouldbe( 1)s−k inthesummandontherightsideoftheseconddisplay
− −
equation, and vice versa for the summand of the right side of the third display equation. At the
bottom of p. 8, Ψ should read Ψ (twice).
n
4
and
1 (2π)2s s t ( 1)t+s
4θ (2s) = 1 − .
3
(cid:18) − 32s−1(cid:19) (22s−1 −1) Xt=1 dXi≥0(cid:18)d1,d2,...,ds(cid:19)3!d15!d2···(2s+1)!ds
Here the sums are such that d +d +...+d = t and d +2d +...+sd = s.
1 2 s 1 2 s
In fact, the minor corner layered determinants of [19, 20] are very special cases of
lower Hessenberg determinants. Following the proof of Theorem 3, we recall a much
more general determinantal result.
Define, for integers a 2 and b 1, the double zeta values
≥ ≥
∞ n−1
1
ζ(a,b) = .
namb
n=1m=1
XX
From manipulating series, it follows that
ζ(a,b)+ζ(b,a) = ζ(a)ζ(b) ζ(a+b).
−
We note that any recurrence of Riemann zeta values (or of Bernoulli numbers) of the
form f(s,j)ζ(2j)ζ(2s 2j) for some function f then leads to a sum formula for
j −
doublPe zeta values. For we have, for instance,
f(s,j)[ζ(2j,2s 2j)+ζ(2s 2j,2j)]= f(s,j)ζ(2j)ζ(2s 2j) ζ(2s) f(s,j).
− − − −
j j j
X X X
As an illustration we have the following.
Theorem 4. For integers s > 1,
s−1 1 1 2−2s−1
1 [ζ(2k,2s 2k)+ζ(2s 2k,2k)] = (4s +6s 1)ζ(2s).
22(s−k) − 22k − − 3 −
k=1 (cid:18) (cid:19)
X
Let E (x) = n n Ek x 1 n−k, with E = 0 and E = 2kE (1/2) denote
n k=0 k 2k − 2 2n+1 k k
the Euler polynoPmial. (cid:0)It(cid:1)has(cid:0)a well(cid:1)known exponential generating function
2ext ∞ tn
= E (x) , t < π. (1.4)
et +1 n n! | |
n=0
X
5
We recall a connection with the alternating zeta function
∞ ( 1)n−1
η(s) = − = (1 21−s)ζ(s).
ns −
n=1
X
We have the evaluation η( j) = ( 1)jE (0)/2 for j 0. The values E (0) =
j n
− − ≥
( 1)nE (1) are 0 for n 2 and E (0) = 1, while otherwise ([1], p. 805) E (0) =
n 0 n
− ≥
2(n + 1)−1(2n+1 1)B for n 1. Many results analogous to Theorem 1 are
n+1
− − ≥
possible, and the following provides a brief example.
Theorem 5. For integers n > 0,
n−1 2(n−m)−1
1 2n 1 E 1
(1 31−2n)(22n 1)B = − 2m .
2n
−2n − − 2m (2m)! −6
m=0(cid:18) (cid:19) (cid:18) (cid:19)
X
Theorem 6. For integers j 0,
≥
j
2 (1 21−2m)(1 22(m−j)+1)ζ(2m)ζ(2j 2m) = (1 2j)ζ(2j).
− − − − −
m=0
X
Theorem 1.3 of [20] gives equivalent forms of the recurrence
s−1
2
ζ(2s+2) = (22k+2 1)ζ(2s 2k)ζ(2k+2). (1.5)
22s+2 1 − −
− k=0
X
Hence, if we introduce functions θ˜ (s) = (1 j−s)ζ(s), we may write
j
−
(22s+2 1) s−1 (22k+2 1)
− θ˜ (2s+2) = 2j2(s+1) − φ (2s 2k)θ˜ (2k +2).
(1 j−(2s+2)) j (j2(k+1) 1) j − j
− k=0 −
X
In fact, (1.5) is proved in [3] (p. 406) as Theorem 3.4. In addition, we may note that
besides the well known relation (e.g., [21, 24])
n−1
2
ζ(2n) = ζ(2k)ζ(2n 2k),
2n+1 −
k=1
X
Williams [24] some time ago proved
n
1
(2k 1) (2n 2k +1) = n (1 2−2n)ζ(2n),
L − L − − 2 −
k=1 (cid:18) (cid:19)
X
6
where is the function
L
∞ ( 1)k 1 1 3
(s) = − = 2−sΦ 1,s, = 4−s ζ s, ζ s, ,
L (2k +1)s − 2 4 − 4
k=0 (cid:18) (cid:19) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
X
and Φ is the Lerch zeta function (see Theorem 9).
We point out that each part of Theorem 1 permits the identification of candidate
pseudo-characteristic polynomials for the Riemann zeta and other related functions.
For instance, if we separate the m = 0 and m = j terms of the sum of the right side
of Theorem 1(a), we obtain
(3j 1) 2π 2j j−1 ( 1)m22m+1π2mζ(2j 2m)
3(3−2j 1)ζ(2j) = ( 1)j − + − − .
− − (2j)! 3 (2m)!32m
(cid:18) (cid:19) m=1
X
We then put
j−1
1 ( 1)m22m+1π2m
p(3)(x) = − x2m
j 3(3−2j 1) (2m)!32m
− m=1
X
and
2j
(3j 1) 2π
z(3)(x) = ( 1)j − +p(3)(x).
j − (2j)!3(3−2j 1) 3 j
− (cid:18) (cid:19)
We recognize that
2 2π ∞ ( 1)m 2π 2m
(3)
p (x) = 1+cos x − x .
j 3(3−2j 1) − 3 − (2m)! 3
" #
− (cid:18) (cid:19) m=j (cid:18) (cid:19)
X
It then appears that inequalities of the following sort hold, where k is either 2s or
2s 1.
−
Conjecture 1. For integers s 4,
≥
ζ(k) ζ(k) 2 z(3)(ζ(k)) ζ(k)+ ζ(k) .
−{ } ≤ s ≤ { }
Here x = x [x] denotes the fractional part of x.
{ } −
We provide new identities for the Ramanujan R (z) and generalized Ramanu-
2s+1
jan R (z) polynomials, and for other functions. For this we require the following
2s
definitions. Let the numbers B∗ for s 2 be given by
s ≥
s−1
1 s+1
B∗ = 2k−sB ,
s −s+1 k k
k=0(cid:18) (cid:19)
X
7
which are then such that B∗ = B and B∗ = (1 2−2s)2B /s. To these values
2s 2s 2s−1 − 2s
are prepended the initial values B∗ = 1 and B∗ = 1/4. The Ramanujan polynomials
0 1
are given by
s+1
B B
R (z) = 2k 2s+2−2k z2k,
2s+1
(2k)!(2s+2 2k)!
k=0 −
X
while the generalized Ramanujan polynomials Q (z) [20] are given by
r
[(r+1)/2] B∗ B∗
Q (z) = r+1−2k 2k z2k.
r
(r+1 2k)!(2k)!
k=0 −
X
The Pochhammer symbol (a) = Γ(a+j)/Γ(a), with Γdenoting theGammafunction,
j
ψ(z) = Γ′(z)/Γ(z) denotes the digamma function, and H = n 1/k the nth
n k=1
harmonic number. P
Theorem 7. (a) The Ramanujan polynomials satisfy the identities
n−1
n (n 1)!
[1 ( 1)n]R(n) (1)+ − ( 1)j(2s+2) R(j) (1)
− − 2s+1 j (j 1)! − − n−j 2s+1
j=1 (cid:18) (cid:19)(cid:20) − (cid:21)
X
= (2s+2) R (1),
n 2s+1
and (b), with R (z) = Q (z), these generalized Ramanujan polynomials satisfy the
2s 2s
identities
1 1
[1 ( 1)n] R(n)(1) R(n)
− − 2s − 2n 2s 2
(cid:20) (cid:18) (cid:19)(cid:21)
n−1
n (n 1)! 1 1
+ − ( 1)j(2s+2) R(j)(1) R(j)
j (j 1)! − − n−j 2s − 2j 2s 2
j=1 (cid:18) (cid:19)(cid:20) − (cid:21)(cid:20) (cid:18) (cid:19)(cid:21)
X
1
= (2s+2) R (1) R .
n 2s 2s
− 2
(cid:20) (cid:18) (cid:19)(cid:21)
(c) Suppose the following functional equation, as appears in Grosswald’s generaliza-
tion [14] of Ramanujan’s formula pertaining to the Riemann zeta function at odd
integer argument ([15], p. 945). For analytic functions F and S,
1 z r z
F ( 1)δ F(z) = S ,
−z − − i i
(cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
8
where z is in the upper half plane and r is real. Then
[( 1)n ( 1)δ]F(n)(i) ( 1)δ( r) inF(i)
n
− − − − − −
n−1
n (n 1)! 1
+ ( 1)n − ( 1)δ( 1)n−j( r) ij−nF(j)(i) = S(n)(1).
j − (j 1)! − − − − n−j in
j=1 (cid:18) (cid:19)(cid:20) − (cid:21)
X
(d) Let F be Zagier’s function [25] (p. 164)
∞ 1 ∞ 1 1
F(x) = [ψ(nx) ln(nx)] = ln(1 e−xt)dt.
n − 1 e−t − t −
n=1 Z0 (cid:18) − (cid:19)
X
Then
1 1 π2 π2 lnx
F′(x) F = + + ,
− x2 x − 6 6x2 x
(cid:18) (cid:19)
and for n 2
≥
n−1 n (n 1)! π2
[1+( 1)n]F(n)(1)+( 1)n − F(j)(1) = ( 1)n−1(n 1)! n H .
n−1
− − j (j 1)! − − 6 −
j=1 (cid:18) (cid:19) − (cid:20) (cid:21)
X
The form of the expressions in (a) and (b) shows that no new information is
included from the even order derivative relations, while the form of (d) shows that no
new information is included for odd n. Explicitly in terms of Bernoulli summations,
the first identity of (a) is the equality of
s+1
B B
2k 2s+2−2k
(s+1)R (1) = (s+1)
2s+1
(2k)!(2s+2 2k)!
k=0 −
X
and
s+1
B B
R′ (1) = 2k 2s+2−2k .
2s+1 (2k 1)!(2s+2 2k)!
k=1 − −
X
Theorem 8. (General Mellin transform result.) Suppose that an analytic func-
tion f has an integral representation
1 ∞
f(s) = ts−1g(t)dt, Re s > 1,
Γ(s)
Z0
for some function g. Putting
√b ∞
cM(b) = g(t)[(√bt 1)2j (√bt+1)2j]dt,
j 2(2j)! − −
Z0
9
we have
j−1 bk
bjf(2j) = cM(b) f(2k).
− j − (2j 2k +1)!
k=1 −
X
Section 3 provides important cases of this result.
In sum, we have generalized the zeta identity of [19], and the Bernoulli and zeta
relations of [20]. We have noted the relevance of a subset of these results for obtaining
summation formulas for double zeta values. In addition, we give an alternative proof
of (1.2) and several other generalizations by using integral representations. Letting-
ton’s identity may then be viewed as a very special case of a recurrence among values
of polygamma functions, and beyond this context, among values of Dirichlet L series,
values of the Lerch zeta function or values of lattice Dirichlet series.
Proof of Theorems
Theorem 1. (a) We recall the expression for Bernoulli polynomials B (x) =
s
s s B xk, so that
k=0 k s−k
j
j
P (cid:0) (cid:1) B (x) = B xj−k.
j k
k
k=0(cid:18) (cid:19)
X
For n = 2j even we have the relation B (1/3) = B (2/3) = (31−n 1)B /2 and
n n n
−
obtain
2j
1 2j 1
(31−2j 1)B = B
2 − 2j k k32j−k
k=0(cid:18) (cid:19)
X
2j
B B 2j 1
0 1
= +2j + B
32j 32j−1 k k32j−k
k=2(cid:18) (cid:19)
X
j
1 j 2j 1
= + B
32j − 32j−1 2m 2m32(j−m)
m=1(cid:18) (cid:19)
X
j
j 2j 1
= + B .
2m
−32j−1 2m 32(j−m)
m=0(cid:18) (cid:19)
X
The relation (1.1) is then applied.
10