Closed forms for generating series, and finite summation analogues - - PowerPoint PPT Presentation
Closed forms for generating series, and finite summation analogues modulo a prime Sandro Mattarei (joint work with Roberto Tauraso) School of Mathematics and Physics University of Lincoln (UK) Gargnano, October 2017 S. Mattarei Finite
Sandro Mattarei (joint work with Roberto Tauraso)
School of Mathematics and Physics University of Lincoln (UK)
Gargnano, October 2017
Finite analogues of generating functions Gargnano, October 2017 1 / 14
Apéry’s celebrated proof (1979) that ζ(3) = ∞
k=1 1/k3 is irrational
relied on the fast converging series
∞
(−1)k k32k
k
= −2
5ζ(3).
Finite analogues of generating functions Gargnano, October 2017 2 / 14
Apéry’s celebrated proof (1979) that ζ(3) = ∞
k=1 1/k3 is irrational
relied on the fast converging series
∞
(−1)k k32k
k
= −2
5ζ(3). Tauraso (2010): for any prime p > 5 we have
p−1
(−1)k k32k
k
≡ 2
5 Hp−1 p2 (mod p3), where Hk = p−1
k=1 1/k denote the harmonic numbers.
Finite analogues of generating functions Gargnano, October 2017 2 / 14
Apéry’s celebrated proof (1979) that ζ(3) = ∞
k=1 1/k3 is irrational
relied on the fast converging series
∞
(−1)k k32k
k
= −2
5ζ(3). Tauraso (2010): for any prime p > 5 we have
p−1
(−1)k k32k
k
≡ 2
5 Hp−1 p2 (mod p3), where Hk = p−1
k=1 1/k denote the harmonic numbers.
Note that p divides
2k
k
for (p − 1)/2 ≤ k < p, but because of
cancellation the LHS turns out to be p-integral.
Finite analogues of generating functions Gargnano, October 2017 2 / 14
Apéry’s celebrated proof (1979) that ζ(3) = ∞
k=1 1/k3 is irrational
relied on the fast converging series
∞
(−1)k k32k
k
= −2
5ζ(3). Tauraso (2010): for any prime p > 5 we have
p−1
(−1)k k32k
k
≡ 2
5 Hp−1 p2 (mod p3), where Hk = p−1
k=1 1/k denote the harmonic numbers.
Note that p divides
2k
k
for (p − 1)/2 ≤ k < p, but because of
cancellation the LHS turns out to be p-integral. Note also that Hp−1 ≡ 0 (mod p2) by Wolstenholme’s theorem.
Finite analogues of generating functions Gargnano, October 2017 2 / 14
Apéry’s celebrated proof (1979) that ζ(3) = ∞
k=1 1/k3 is irrational
relied on the fast converging series
∞
(−1)k k32k
k
= −2
5ζ(3). Tauraso (2010): for any prime p > 5 we have
p−1
(−1)k k32k
k
≡ 2
5 Hp−1 p2 (mod p3), where Hk = p−1
k=1 1/k denote the harmonic numbers.
Note that p divides
2k
k
for (p − 1)/2 ≤ k < p, but because of
cancellation the LHS turns out to be p-integral. Note also that Hp−1 ≡ 0 (mod p2) by Wolstenholme’s theorem. There is a clear similarity in the coefficient 2/5, but the rest of the analogy is more mysterious.
Finite analogues of generating functions Gargnano, October 2017 2 / 14
∞
(4k + 1)
k
3
= 2 π (Ramanujan)
Finite analogues of generating functions Gargnano, October 2017 3 / 14
∞
(4k + 1)
k
3
= 2 π (Ramanujan) Note that
k
k
Finite analogues of generating functions Gargnano, October 2017 3 / 14
∞
(4k + 1)
k
3
= 2 π (Ramanujan) Note that
k
k
Van Hamme (1996) conjectured, and Mortenson (2008) proved
p−1
(4k + 1)
k
3
≡
−1
p
for p > 2, where
p
Finite analogues of generating functions Gargnano, October 2017 3 / 14
∞
(4k + 1)
k
3
= 2 π (Ramanujan) Note that
k
k
Van Hamme (1996) conjectured, and Mortenson (2008) proved
p−1
(4k + 1)
k
3
≡
−1
p
for p > 2, where
p
One can only see the analogy after re-writing the right-hand sides in terms of values of the gamma function, and of the p-adic gamma function, each evaluated at 1/2 in this case.
Finite analogues of generating functions Gargnano, October 2017 3 / 14
Some numerical series are specializations of power series in an indeterminate which admit a closed form. Not so for the series used by Apéry, or that of Ramanujan, but for example,
∞
1 k22k
k
= π2
18
Finite analogues of generating functions Gargnano, October 2017 4 / 14
Some numerical series are specializations of power series in an indeterminate which admit a closed form. Not so for the series used by Apéry, or that of Ramanujan, but for example,
∞
1 k22k
k
= π2
18 comes from
∞
xk k22k
k
= 2
√x
2
2
Finite analogues of generating functions Gargnano, October 2017 4 / 14
Some numerical series are specializations of power series in an indeterminate which admit a closed form. Not so for the series used by Apéry, or that of Ramanujan, but for example,
∞
1 k22k
k
= π2
18 comes from
∞
xk k22k
k
= 2
√x
2
2
Mattarei/Tauraso (2013): This has a finite analogue mod p2: p
p−1
xk k22k
k
≡ 2 − αp − α−p − xp
2p (mod p2), where α2 + (x − 2)α + 1 = 0, and one more complicated mod p3.
Finite analogues of generating functions Gargnano, October 2017 4 / 14
Some numerical series are specializations of power series in an indeterminate which admit a closed form. Not so for the series used by Apéry, or that of Ramanujan, but for example,
∞
1 k22k
k
= π2
18 comes from
∞
xk k22k
k
= 2
√x
2
2
Mattarei/Tauraso (2013): This has a finite analogue mod p2: p
p−1
xk k22k
k
≡ 2 − αp − α−p − xp
2p (mod p2), where α2 + (x − 2)α + 1 = 0, and one more complicated mod p3. Hard to see a similarity in this case.
Finite analogues of generating functions Gargnano, October 2017 4 / 14
We look at a class of power series having a closed form, which have a finite analogue mod p, also with a closed form.
Finite analogues of generating functions Gargnano, October 2017 5 / 14
We look at a class of power series having a closed form, which have a finite analogue mod p, also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former.
Finite analogues of generating functions Gargnano, October 2017 5 / 14
We look at a class of power series having a closed form, which have a finite analogue mod p, also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. Let q be a power of a prime p. Shorten ‘≡ (mod p)’ to ‘≡p’.
Finite analogues of generating functions Gargnano, October 2017 5 / 14
We look at a class of power series having a closed form, which have a finite analogue mod p, also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. Let q be a power of a prime p. Shorten ‘≡ (mod p)’ to ‘≡p’. Conditions such as p > 2 will be omitted for simplicity.
Finite analogues of generating functions Gargnano, October 2017 5 / 14
We look at a class of power series having a closed form, which have a finite analogue mod p, also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. Let q be a power of a prime p. Shorten ‘≡ (mod p)’ to ‘≡p’. Conditions such as p > 2 will be omitted for simplicity. Here is a simple example.
∞
k
1 √ 1 − 4x ⇒
q−1
k
Finite analogues of generating functions Gargnano, October 2017 5 / 14
We look at a class of power series having a closed form, which have a finite analogue mod p, also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. Let q be a power of a prime p. Shorten ‘≡ (mod p)’ to ‘≡p’. Conditions such as p > 2 will be omitted for simplicity. Here is a simple example.
∞
k
1 √ 1 − 4x ⇒
q−1
k
Two basic tricks are available for this deduction:
Finite analogues of generating functions Gargnano, October 2017 5 / 14
We look at a class of power series having a closed form, which have a finite analogue mod p, also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. Let q be a power of a prime p. Shorten ‘≡ (mod p)’ to ‘≡p’. Conditions such as p > 2 will be omitted for simplicity. Here is a simple example.
∞
k
1 √ 1 − 4x ⇒
q−1
k
Two basic tricks are available for this deduction:
◮ (1 + x)q ≡p 1 + xq;
Finite analogues of generating functions Gargnano, October 2017 5 / 14
We look at a class of power series having a closed form, which have a finite analogue mod p, also with a closed form. Our goal is not just finding the latter, but deducing it directly from the former. Let q be a power of a prime p. Shorten ‘≡ (mod p)’ to ‘≡p’. Conditions such as p > 2 will be omitted for simplicity. Here is a simple example.
∞
k
1 √ 1 − 4x ⇒
q−1
k
Two basic tricks are available for this deduction:
◮ (1 + x)q ≡p 1 + xq; ◮ if A(x) ≡ B(x) (mod C(x)), for polynomials over a field, and
deg A, deg B < deg C, then A(x) = B(x).
Finite analogues of generating functions Gargnano, October 2017 5 / 14
First use (1 + x)q ≡p 1 + xq:
∞
k
= (1 − 4x)(q−1)/2 ·
(1 − 4x)q−1/2
≡p (1 − 4x)(q−1)/2 · (1 − 4xq)−1/2 ≡(p,xq) (1 − 4x)(q−1)/2
Finite analogues of generating functions Gargnano, October 2017 6 / 14
First use (1 + x)q ≡p 1 + xq:
∞
k
= (1 − 4x)(q−1)/2 ·
(1 − 4x)q−1/2
≡p (1 − 4x)(q−1)/2 · (1 − 4xq)−1/2 ≡(p,xq) (1 − 4x)(q−1)/2 The notation ≡(p,xq) here might mean equality in Z[x]/(p, xq).
Finite analogues of generating functions Gargnano, October 2017 6 / 14
First use (1 + x)q ≡p 1 + xq:
∞
k
= (1 − 4x)(q−1)/2 ·
(1 − 4x)q−1/2
≡p (1 − 4x)(q−1)/2 · (1 − 4xq)−1/2 ≡(p,xq) (1 − 4x)(q−1)/2 The notation ≡(p,xq) here might mean equality in Z[x]/(p, xq). However, better read it as first ignore the terms of degree ≥ q, and
all coefficients are p-integral, such as that for log(1 + x).
Finite analogues of generating functions Gargnano, October 2017 6 / 14
First use (1 + x)q ≡p 1 + xq:
∞
k
= (1 − 4x)(q−1)/2 ·
(1 − 4x)q−1/2
≡p (1 − 4x)(q−1)/2 · (1 − 4xq)−1/2 ≡(p,xq) (1 − 4x)(q−1)/2 The notation ≡(p,xq) here might mean equality in Z[x]/(p, xq). However, better read it as first ignore the terms of degree ≥ q, and
all coefficients are p-integral, such as that for log(1 + x). Now because ∞
k=0
2k
k
xk and (1 − 4x)(q−1)/2 have each degree
< q, the condition ≡(p,xq) implies ≡p.
Finite analogues of generating functions Gargnano, October 2017 6 / 14
We set β = ∞
k=0 Ckxk+1 = 1− √ 1−4x 2
, the generating function of the Catalan numbers Ck =
1 k+1
2k
k
. Note that x = β(1 − β).
Finite analogues of generating functions Gargnano, October 2017 7 / 14
We set β = ∞
k=0 Ckxk+1 = 1− √ 1−4x 2
, the generating function of the Catalan numbers Ck =
1 k+1
2k
k
. Note that x = β(1 − β).
∞
k
1 (1 − 2β)(1 − β)e
Finite analogues of generating functions Gargnano, October 2017 7 / 14
We set β = ∞
k=0 Ckxk+1 = 1− √ 1−4x 2
, the generating function of the Catalan numbers Ck =
1 k+1
2k
k
. Note that x = β(1 − β).
∞
k
1 (1 − 2β)(1 − β)e One can view this as an identity in Q[[x]], but better view it as an identity in Q[[β]], where x = β(1 − β). Also write α = 1 − β.
Finite analogues of generating functions Gargnano, October 2017 7 / 14
We set β = ∞
k=0 Ckxk+1 = 1− √ 1−4x 2
, the generating function of the Catalan numbers Ck =
1 k+1
2k
k
. Note that x = β(1 − β).
∞
k
1 (1 − 2β)(1 − β)e One can view this as an identity in Q[[x]], but better view it as an identity in Q[[β]], where x = β(1 − β). Also write α = 1 − β. ⇒
q−1
k
α2q−e − β2q−e α − β
Finite analogues of generating functions Gargnano, October 2017 7 / 14
We set β = ∞
k=0 Ckxk+1 = 1− √ 1−4x 2
, the generating function of the Catalan numbers Ck =
1 k+1
2k
k
. Note that x = β(1 − β).
∞
k
1 (1 − 2β)(1 − β)e One can view this as an identity in Q[[x]], but better view it as an identity in Q[[β]], where x = β(1 − β). Also write α = 1 − β. ⇒
q−1
k
α2q−e − β2q−e α − β The RHS is a polynomial in β, so this identity takes place in Fp[β].
Finite analogues of generating functions Gargnano, October 2017 7 / 14
We set β = ∞
k=0 Ckxk+1 = 1− √ 1−4x 2
, the generating function of the Catalan numbers Ck =
1 k+1
2k
k
. Note that x = β(1 − β).
∞
k
1 (1 − 2β)(1 − β)e One can view this as an identity in Q[[x]], but better view it as an identity in Q[[β]], where x = β(1 − β). Also write α = 1 − β. ⇒
q−1
k
α2q−e − β2q−e α − β The RHS is a polynomial in β, so this identity takes place in Fp[β]. The substitution β → 1 − β does not make sense in Q[[β]] (power series), but it does in Fp[β] (polynomials). Hence congruences have an additional symmetry, absent from the series identities.
Finite analogues of generating functions Gargnano, October 2017 7 / 14
∞
k
k = −2 log(1 − β) = 2 Li1 (β)
Finite analogues of generating functions Gargnano, October 2017 8 / 14
∞
k
k = −2 log(1 − β) = 2 Li1 (β) Repeated integration introduces polylogarithms: Lid(x) =
∞
xk kd .
Finite analogues of generating functions Gargnano, October 2017 8 / 14
∞
k
k = −2 log(1 − β) = 2 Li1 (β) Repeated integration introduces polylogarithms: Lid(x) =
∞
xk kd . They have finite analogues: £d(x) =
p−1
xk kd .
Finite analogues of generating functions Gargnano, October 2017 8 / 14
∞
k
k = −2 log(1 − β) = 2 Li1 (β) Repeated integration introduces polylogarithms: Lid(x) =
∞
xk kd . They have finite analogues: £d(x) =
p−1
xk kd . ⇒
p−1
k
k ≡p £1(β) + £1(1 − β)
Finite analogues of generating functions Gargnano, October 2017 8 / 14
∞
k
k = −2 log(1 − β) = 2 Li1 (β) Repeated integration introduces polylogarithms: Lid(x) =
∞
xk kd . They have finite analogues: £d(x) =
p−1
xk kd . ⇒
p−1
k
k ≡p £1(β) + £1(1 − β) Actually, £1(β) = £1(1 − β) (see later on), but we have kept them separate to highlight invariance under β → 1 − β.
Finite analogues of generating functions Gargnano, October 2017 8 / 14
∞
k
k = −2 log(1 − β) = 2 Li1 (β) Repeated integration introduces polylogarithms: Lid(x) =
∞
xk kd . They have finite analogues: £d(x) =
p−1
xk kd . ⇒
p−1
k
k ≡p £1(β) + £1(1 − β) Actually, £1(β) = £1(1 − β) (see later on), but we have kept them separate to highlight invariance under β → 1 − β. More generally, we look at series
k
2k
k
akxk, where ak might be
1/kd, or H(d)
k , or a linear combination of products of them.
Finite analogues of generating functions Gargnano, October 2017 8 / 14
∞
k
k = −2 log(1 − β) = 2 Li1 (β) Repeated integration introduces polylogarithms: Lid(x) =
∞
xk kd . They have finite analogues: £d(x) =
p−1
xk kd . ⇒
p−1
k
k ≡p £1(β) + £1(1 − β) Actually, £1(β) = £1(1 − β) (see later on), but we have kept them separate to highlight invariance under β → 1 − β. More generally, we look at series
k
2k
k
akxk, where ak might be
1/kd, or H(d)
k , or a linear combination of products of them.
We sort them by their level: the highest exponent of k occurring in the denominator of ak once expanded.
Finite analogues of generating functions Gargnano, October 2017 8 / 14
H(d)
k
= k
j=1 1/jd denotes a generalized harmonic number.
Finite analogues of generating functions Gargnano, October 2017 9 / 14
H(d)
k
= k
j=1 1/jd denotes a generalized harmonic number. ∞
k
1 − 2β = −2 1 − 2β Li1
2β − 1
Finite analogues of generating functions Gargnano, October 2017 9 / 14
H(d)
k
= k
j=1 1/jd denotes a generalized harmonic number. ∞
k
1 − 2β = −2 1 − 2β Li1
2β − 1
log(1 + x + y + xy) = log(1 + x) log(1 + y).
Finite analogues of generating functions Gargnano, October 2017 9 / 14
H(d)
k
= k
j=1 1/jd denotes a generalized harmonic number. ∞
k
1 − 2β = −2 1 − 2β Li1
2β − 1
log(1 + x + y + xy) = log(1 + x) log(1 + y). An involutory transform turns a series into another one of the same level. The above formula comes from that for ∞
k=1
2k
k
xk/k.
Finite analogues of generating functions Gargnano, October 2017 9 / 14
H(d)
k
= k
j=1 1/jd denotes a generalized harmonic number. ∞
k
1 − 2β = −2 1 − 2β Li1
2β − 1
log(1 + x + y + xy) = log(1 + x) log(1 + y). An involutory transform turns a series into another one of the same level. The above formula comes from that for ∞
k=1
2k
k
xk/k.
⇒
p−1
k
β − α
Finite analogues of generating functions Gargnano, October 2017 9 / 14
H(d)
k
= k
j=1 1/jd denotes a generalized harmonic number. ∞
k
1 − 2β = −2 1 − 2β Li1
2β − 1
log(1 + x + y + xy) = log(1 + x) log(1 + y). An involutory transform turns a series into another one of the same level. The above formula comes from that for ∞
k=1
2k
k
xk/k.
⇒
p−1
k
β − α
Finite analogues of generating functions Gargnano, October 2017 9 / 14
H(d)
k
= k
j=1 1/jd denotes a generalized harmonic number. ∞
k
1 − 2β = −2 1 − 2β Li1
2β − 1
log(1 + x + y + xy) = log(1 + x) log(1 + y). An involutory transform turns a series into another one of the same level. The above formula comes from that for ∞
k=1
2k
k
xk/k.
⇒
p−1
k
β − α
The RHS is invariant under β → α = 1 − β (as it should since the LHS is) because
α α−β + β β−α = 1 and £1(y) = £1(1 − y).
Finite analogues of generating functions Gargnano, October 2017 9 / 14
∞
k
k2 = 2 Li2(β) − Li1(β)2 ⇒
p−1
k
k2 ≡p 2£2(β) + 2£2(1 − β)
Finite analogues of generating functions Gargnano, October 2017 10 / 14
∞
k
k2 = 2 Li2(β) − Li1(β)2 ⇒
p−1
k
k2 ≡p 2£2(β) + 2£2(1 − β) The term 2£2(β) in the congruence clearly matches 2 Li2(β).
Finite analogues of generating functions Gargnano, October 2017 10 / 14
∞
k
k2 = 2 Li2(β) − Li1(β)2 ⇒
p−1
k
k2 ≡p 2£2(β) + 2£2(1 − β) The term 2£2(β) in the congruence clearly matches 2 Li2(β). Invariance under β → 1 − β requires a term 2£2(1 − β).
Finite analogues of generating functions Gargnano, October 2017 10 / 14
∞
k
k2 = 2 Li2(β) − Li1(β)2 ⇒
p−1
k
k2 ≡p 2£2(β) + 2£2(1 − β) The term 2£2(β) in the congruence clearly matches 2 Li2(β). Invariance under β → 1 − β requires a term 2£2(1 − β). But where has the term − Li1(β)2 gone?
Finite analogues of generating functions Gargnano, October 2017 10 / 14
∞
k
k2 = 2 Li2(β) − Li1(β)2 ⇒
p−1
k
k2 ≡p 2£2(β) + 2£2(1 − β) The term 2£2(β) in the congruence clearly matches 2 Li2(β). Invariance under β → 1 − β requires a term 2£2(1 − β). But where has the term − Li1(β)2 gone? Expanding Li1(x)d one finds £1(x)d ≡(xp,p) (−1)d−1d! · £d(1 − x).
Finite analogues of generating functions Gargnano, October 2017 10 / 14
∞
k
k2 = 2 Li2(β) − Li1(β)2 ⇒
p−1
k
k2 ≡p 2£2(β) + 2£2(1 − β) The term 2£2(β) in the congruence clearly matches 2 Li2(β). Invariance under β → 1 − β requires a term 2£2(1 − β). But where has the term − Li1(β)2 gone? Expanding Li1(x)d one finds £1(x)d ≡(xp,p) (−1)d−1d! · £d(1 − x). Sharper versions with ‘≡p’ are only known for d = 1, 2, 3, since Mirimanoff (∼1900), but rediscovered recently by several authors.
Finite analogues of generating functions Gargnano, October 2017 10 / 14
∞
k
k2 = 2 Li2(β) − Li1(β)2 ⇒
p−1
k
k2 ≡p 2£2(β) + 2£2(1 − β) The term 2£2(β) in the congruence clearly matches 2 Li2(β). Invariance under β → 1 − β requires a term 2£2(1 − β). But where has the term − Li1(β)2 gone? Expanding Li1(x)d one finds £1(x)d ≡(xp,p) (−1)d−1d! · £d(1 − x). Sharper versions with ‘≡p’ are only known for d = 1, 2, 3, since Mirimanoff (∼1900), but rediscovered recently by several authors. £1(x) ≡p £1(1 − x), follows straight from £1(x) ≡p
−xp−(1−x)p+1 p
.
Finite analogues of generating functions Gargnano, October 2017 10 / 14
∞
k
k2 = 2 Li2(β) − Li1(β)2 ⇒
p−1
k
k2 ≡p 2£2(β) + 2£2(1 − β) The term 2£2(β) in the congruence clearly matches 2 Li2(β). Invariance under β → 1 − β requires a term 2£2(1 − β). But where has the term − Li1(β)2 gone? Expanding Li1(x)d one finds £1(x)d ≡(xp,p) (−1)d−1d! · £d(1 − x). Sharper versions with ‘≡p’ are only known for d = 1, 2, 3, since Mirimanoff (∼1900), but rediscovered recently by several authors. £1(x) ≡p £1(1 − x), follows straight from £1(x) ≡p
−xp−(1−x)p+1 p
. £1(x)2/2 ≡p −xp£2(x) − (1 − xp)£2(1 − x).
Finite analogues of generating functions Gargnano, October 2017 10 / 14
£1(x)3/6 ≡p xp£3(x) + (1 − xp)£3(1 − x) + x2p(1 − xp)£3(1 − 1/x) +(2/3)xp(1 − xp)£3(−1).
Finite analogues of generating functions Gargnano, October 2017 11 / 14
£1(x)3/6 ≡p xp£3(x) + (1 − xp)£3(1 − x) + x2p(1 − xp)£3(1 − 1/x) +(2/3)xp(1 − xp)£3(−1). This allows us to deal with the following series of level three:
∞
k H(2)
k
k + 1 k3
3 Li1(β)3 ⇒
p−1
k H(2)
k
k + 1 k3
Finite analogues of generating functions Gargnano, October 2017 11 / 14
£1(x)3/6 ≡p xp£3(x) + (1 − xp)£3(1 − x) + x2p(1 − xp)£3(1 − 1/x) +(2/3)xp(1 − xp)£3(−1). This allows us to deal with the following series of level three:
∞
k H(2)
k
k + 1 k3
3 Li1(β)3 ⇒
p−1
k H(2)
k
k + 1 k3
The summands H(2)
k /k and 1/k3 cannot be separated for the
series, but they can for the congruence:
p−1
k
k
k xk ≡p −2xp(£3(1 − 1/α) + £3(1 − 1/β))
Finite analogues of generating functions Gargnano, October 2017 11 / 14
£1(x)3/6 ≡p xp£3(x) + (1 − xp)£3(1 − x) + x2p(1 − xp)£3(1 − 1/x) +(2/3)xp(1 − xp)£3(−1). This allows us to deal with the following series of level three:
∞
k H(2)
k
k + 1 k3
3 Li1(β)3 ⇒
p−1
k H(2)
k
k + 1 k3
The summands H(2)
k /k and 1/k3 cannot be separated for the
series, but they can for the congruence:
p−1
k
k
k xk ≡p −2xp(£3(1 − 1/α) + £3(1 − 1/β)) More success in the modular case is due to extra symmetries.
Finite analogues of generating functions Gargnano, October 2017 11 / 14
The above congruences for £1(x)2 and £1(x)3 are themselves given new proofs which deduce them from truncating Li1(x)2 and Li1(x)3 by exploiting symmetries (functional equations mod p).
Finite analogues of generating functions Gargnano, October 2017 12 / 14
The above congruences for £1(x)2 and £1(x)3 are themselves given new proofs which deduce them from truncating Li1(x)2 and Li1(x)3 by exploiting symmetries (functional equations mod p). For example, because £1(x) ≡p −xp£1(1/x), the lower half of the coefficients of £1(x)2 ≡p x2p£1(1/x)2 determines the upper half. But the lower half can be found by inspecting Li1(x)2.
Finite analogues of generating functions Gargnano, October 2017 12 / 14
The above congruences for £1(x)2 and £1(x)3 are themselves given new proofs which deduce them from truncating Li1(x)2 and Li1(x)3 by exploiting symmetries (functional equations mod p). For example, because £1(x) ≡p −xp£1(1/x), the lower half of the coefficients of £1(x)2 ≡p x2p£1(1/x)2 determines the upper half. But the lower half can be found by inspecting Li1(x)2. The symmetries £1(1 − x) = £1(x) and £1(x) ≡p −xp£1(1/x) together generate a group G ∼ = S3. One deduces that all the coefficients of £1(x)3 (except for two) are determined by its lowest third of coefficients, which can be found from Li1(x)3.
Finite analogues of generating functions Gargnano, October 2017 12 / 14
The above congruences for £1(x)2 and £1(x)3 are themselves given new proofs which deduce them from truncating Li1(x)2 and Li1(x)3 by exploiting symmetries (functional equations mod p). For example, because £1(x) ≡p −xp£1(1/x), the lower half of the coefficients of £1(x)2 ≡p x2p£1(1/x)2 determines the upper half. But the lower half can be found by inspecting Li1(x)2. The symmetries £1(1 − x) = £1(x) and £1(x) ≡p −xp£1(1/x) together generate a group G ∼ = S3. One deduces that all the coefficients of £1(x)3 (except for two) are determined by its lowest third of coefficients, which can be found from Li1(x)3. If the full group G is exploited for £1(x)2, one also deduces the functional equation £2(x) ≡p £2(1 − x) + xp£2(1 − 1/x).
Finite analogues of generating functions Gargnano, October 2017 12 / 14
The above congruences for £1(x)2 and £1(x)3 are themselves given new proofs which deduce them from truncating Li1(x)2 and Li1(x)3 by exploiting symmetries (functional equations mod p). For example, because £1(x) ≡p −xp£1(1/x), the lower half of the coefficients of £1(x)2 ≡p x2p£1(1/x)2 determines the upper half. But the lower half can be found by inspecting Li1(x)2. The symmetries £1(1 − x) = £1(x) and £1(x) ≡p −xp£1(1/x) together generate a group G ∼ = S3. One deduces that all the coefficients of £1(x)3 (except for two) are determined by its lowest third of coefficients, which can be found from Li1(x)3. If the full group G is exploited for £1(x)2, one also deduces the functional equation £2(x) ≡p £2(1 − x) + xp£2(1 − 1/x). Functional equations for (finite) polylogarithms allow calculation of certain special values, and obtain specializations such as
p−1
k
k3 ≡p 2Bp−3 3 , where Bp−3 is a Bernoulli number.
Finite analogues of generating functions Gargnano, October 2017 12 / 14
£1(x) − £1(y) + xp£1
y
x
1 − y
1 − x
Finite analogues of generating functions Gargnano, October 2017 13 / 14
£1(x) − £1(y) + xp£1
y
x
1 − y
1 − x
◮ Note that this takes place in Fp[x, y] after cancellation.
Also, the LHS has (apparent) degree at most p.
Finite analogues of generating functions Gargnano, October 2017 13 / 14
£1(x) − £1(y) + xp£1
y
x
1 − y
1 − x
◮ Note that this takes place in Fp[x, y] after cancellation.
Also, the LHS has (apparent) degree at most p.
◮ Discussed by Kontsevich (1992), asking for analogues for £2.
Finite analogues of generating functions Gargnano, October 2017 13 / 14
£1(x) − £1(y) + xp£1
y
x
1 − y
1 − x
◮ Note that this takes place in Fp[x, y] after cancellation.
Also, the LHS has (apparent) degree at most p.
◮ Discussed by Kontsevich (1992), asking for analogues for £2. ◮ Elbaz-Vincent and Gangl (2002) developed a way of deducing
functional equations for £d from functional equations for Lid+1.
Finite analogues of generating functions Gargnano, October 2017 13 / 14
£1(x) − £1(y) + xp£1
y
x
1 − y
1 − x
◮ Note that this takes place in Fp[x, y] after cancellation.
Also, the LHS has (apparent) degree at most p.
◮ Discussed by Kontsevich (1992), asking for analogues for £2. ◮ Elbaz-Vincent and Gangl (2002) developed a way of deducing
functional equations for £d from functional equations for Lid+1.
However, here is a new derivation of the above, from the functional equation for Li1: − log(1 − x) + log(1 − y) = log
1−y
1−x
Finite analogues of generating functions Gargnano, October 2017 13 / 14
£1(x) − £1(y) + xp£1
y
x
1 − y
1 − x
◮ Note that this takes place in Fp[x, y] after cancellation.
Also, the LHS has (apparent) degree at most p.
◮ Discussed by Kontsevich (1992), asking for analogues for £2. ◮ Elbaz-Vincent and Gangl (2002) developed a way of deducing
functional equations for £d from functional equations for Lid+1.
However, here is a new derivation of the above, from the functional equation for Li1: − log(1 − x) + log(1 − y) = log
1−y
1−x
◮ Truncation gives £1(x) − £1(y) + £1
1 − x
congruence being modulo ((x, y)p, p) in Zp[[x, y]], so accounting for all terms of degree < p in the desired congruence.
Finite analogues of generating functions Gargnano, October 2017 13 / 14
£1(x) − £1(y) + xp£1
y
x
1 − y
1 − x
◮ Note that this takes place in Fp[x, y] after cancellation.
Also, the LHS has (apparent) degree at most p.
◮ Discussed by Kontsevich (1992), asking for analogues for £2. ◮ Elbaz-Vincent and Gangl (2002) developed a way of deducing
functional equations for £d from functional equations for Lid+1.
However, here is a new derivation of the above, from the functional equation for Li1: − log(1 − x) + log(1 − y) = log
1−y
1−x
◮ Truncation gives £1(x) − £1(y) + £1
1 − x
congruence being modulo ((x, y)p, p) in Zp[[x, y]], so accounting for all terms of degree < p in the desired congruence.
◮ Invariance under the group G of symmetries produces the
additional terms, and also resolves the residual indeterminacy about the terms of degree p.
Finite analogues of generating functions Gargnano, October 2017 13 / 14
Finite analogues of generating functions Gargnano, October 2017 14 / 14