non-holonomic sequences and functions
TRANSCRIPT
Non-Holonomic Sequences and Functions
Stefan Gerhold
May 16, 2006
Holonomic Sequences and Functions
I A function f (z) is holonomic if it satisfies an LODE
p0(z)f (0)(z) + · · ·+ pd(z)f (d)(z) = 0
with polynomial coefficients.
I A sequence (an) is holonomic if it satisfies an LORE
p0(n)an + · · ·+ pd(n)an+d = 0
with polynomial coefficients.
I Rather general way to specify concrete functions/sequences infinite terms
I Several algorithms are available for the symbolic manipulationof holonomic functions and sequences.
Holonomic Sequences and Functions
I A function f (z) is holonomic if it satisfies an LODE
p0(z)f (0)(z) + · · ·+ pd(z)f (d)(z) = 0
with polynomial coefficients.
I A sequence (an) is holonomic if it satisfies an LORE
p0(n)an + · · ·+ pd(n)an+d = 0
with polynomial coefficients.
I Rather general way to specify concrete functions/sequences infinite terms
I Several algorithms are available for the symbolic manipulationof holonomic functions and sequences.
Holonomic Sequences and Functions
I A function f (z) is holonomic if it satisfies an LODE
p0(z)f (0)(z) + · · ·+ pd(z)f (d)(z) = 0
with polynomial coefficients.
I A sequence (an) is holonomic if it satisfies an LORE
p0(n)an + · · ·+ pd(n)an+d = 0
with polynomial coefficients.
I Rather general way to specify concrete functions/sequences infinite terms
I Several algorithms are available for the symbolic manipulationof holonomic functions and sequences.
Holonomic Sequences and Functions
I A function f (z) is holonomic if it satisfies an LODE
p0(z)f (0)(z) + · · ·+ pd(z)f (d)(z) = 0
with polynomial coefficients.
I A sequence (an) is holonomic if it satisfies an LORE
p0(n)an + · · ·+ pd(n)an+d = 0
with polynomial coefficients.
I Rather general way to specify concrete functions/sequences infinite terms
I Several algorithms are available for the symbolic manipulationof holonomic functions and sequences.
Examples and Basic Properties
I Examples of holonomic functions: rational functions, exp, sin,cos, Bessel, etc.
I Examples of holonomic sequences: αn, n!,∑n
k=1 1/k,](involutions), etc.
I Sum and Product of two holonomic sequences (functions) areholonomic
I an is holonomic iff∑
n≥0 anzn is holonomic
I Definition can be extended to several variables(n1, . . . , nr , z1, . . . , zs)
Examples and Basic Properties
I Examples of holonomic functions: rational functions, exp, sin,cos, Bessel, etc.
I Examples of holonomic sequences: αn, n!,∑n
k=1 1/k,](involutions), etc.
I Sum and Product of two holonomic sequences (functions) areholonomic
I an is holonomic iff∑
n≥0 anzn is holonomic
I Definition can be extended to several variables(n1, . . . , nr , z1, . . . , zs)
Examples and Basic Properties
I Examples of holonomic functions: rational functions, exp, sin,cos, Bessel, etc.
I Examples of holonomic sequences: αn, n!,∑n
k=1 1/k,](involutions), etc.
I Sum and Product of two holonomic sequences (functions) areholonomic
I an is holonomic iff∑
n≥0 anzn is holonomic
I Definition can be extended to several variables(n1, . . . , nr , z1, . . . , zs)
Examples and Basic Properties
I Examples of holonomic functions: rational functions, exp, sin,cos, Bessel, etc.
I Examples of holonomic sequences: αn, n!,∑n
k=1 1/k,](involutions), etc.
I Sum and Product of two holonomic sequences (functions) areholonomic
I an is holonomic iff∑
n≥0 anzn is holonomic
I Definition can be extended to several variables(n1, . . . , nr , z1, . . . , zs)
Examples and Basic Properties
I Examples of holonomic functions: rational functions, exp, sin,cos, Bessel, etc.
I Examples of holonomic sequences: αn, n!,∑n
k=1 1/k,](involutions), etc.
I Sum and Product of two holonomic sequences (functions) areholonomic
I an is holonomic iff∑
n≥0 anzn is holonomic
I Definition can be extended to several variables(n1, . . . , nr , z1, . . . , zs)
Proving Non-Holonomicity
I Holonomic sequences ≈ algebraic numbers
I Annihilated by operators instead of polynomials
I If a sequence (function) is not obviously holonomic, it isusually not holonomic
I But how to come up with a rigorous proof?
Proving Non-Holonomicity
I Holonomic sequences ≈ algebraic numbers
I Annihilated by operators instead of polynomials
I If a sequence (function) is not obviously holonomic, it isusually not holonomic
I But how to come up with a rigorous proof?
Proving Non-Holonomicity
I Holonomic sequences ≈ algebraic numbers
I Annihilated by operators instead of polynomials
I If a sequence (function) is not obviously holonomic, it isusually not holonomic
I But how to come up with a rigorous proof?
Why?
I Lower bound on the algorithmic complexity of a sequence(function)
I Relations to many areas of mathematics:
I Analytic combinatorics, complex analysis, algebraic geometry,number theory
I Many ways to define sequences (functions) =⇒ ampleopportunities for research
Why?
I Lower bound on the algorithmic complexity of a sequence(function)
I Relations to many areas of mathematics:
I Analytic combinatorics, complex analysis, algebraic geometry,number theory
I Many ways to define sequences (functions) =⇒ ampleopportunities for research
Why?
I Lower bound on the algorithmic complexity of a sequence(function)
I Relations to many areas of mathematics:
I Analytic combinatorics, complex analysis, algebraic geometry,number theory
I Many ways to define sequences (functions) =⇒ ampleopportunities for research
Two Easy arguments
I For each holonomic sequence an there is α such thatan = O(n!α)
I Hence 22nand 2n2
are not holonomic (What about nn ?).
I Other argument: Singularities of a holonomic function f (z)are roots of pd(z).
I Hence tan z , z/(ez −1), and∏
(1− zn)−1 are not holonomic.
Two Easy arguments
I For each holonomic sequence an there is α such thatan = O(n!α)
I Hence 22nand 2n2
are not holonomic (What about nn ?).
I Other argument: Singularities of a holonomic function f (z)are roots of pd(z).
I Hence tan z , z/(ez −1), and∏
(1− zn)−1 are not holonomic.
Two Easy arguments
I For each holonomic sequence an there is α such thatan = O(n!α)
I Hence 22nand 2n2
are not holonomic (What about nn ?).
I Other argument: Singularities of a holonomic function f (z)are roots of pd(z).
I Hence tan z , z/(ez −1), and∏
(1− zn)−1 are not holonomic.
Two Easy arguments
I For each holonomic sequence an there is α such thatan = O(n!α)
I Hence 22nand 2n2
are not holonomic (What about nn ?).
I Other argument: Singularities of a holonomic function f (z)are roots of pd(z).
I Hence tan z , z/(ez −1), and∏
(1− zn)−1 are not holonomic.
Infinitely Many Singularities
I We show that the sequence (ζ(n))n≥2 of values of theRiemann zeta function is not holonomic.
I
feven(z) :=∑n≥1
ζ(2n)zn, fodd(z) :=∑n≥1
ζ(2n + 1)zn
I Euler:feven(z) = 1
2 −π2 z cotπz
I Molteni (2001):
feven(z)/z − fodd(z) = γ − ψ(1 + z),
where ψ(z) = Γ′(z)/Γ(z) is the digamma function.
Infinitely Many Singularities
I We show that the sequence (ζ(n))n≥2 of values of theRiemann zeta function is not holonomic.
I
feven(z) :=∑n≥1
ζ(2n)zn, fodd(z) :=∑n≥1
ζ(2n + 1)zn
I Euler:feven(z) = 1
2 −π2 z cotπz
I Molteni (2001):
feven(z)/z − fodd(z) = γ − ψ(1 + z),
where ψ(z) = Γ′(z)/Γ(z) is the digamma function.
Infinitely Many Singularities
I We show that the sequence (ζ(n))n≥2 of values of theRiemann zeta function is not holonomic.
I
feven(z) :=∑n≥1
ζ(2n)zn, fodd(z) :=∑n≥1
ζ(2n + 1)zn
I Euler:feven(z) = 1
2 −π2 z cotπz
I Molteni (2001):
feven(z)/z − fodd(z) = γ − ψ(1 + z),
where ψ(z) = Γ′(z)/Γ(z) is the digamma function.
Infinitely Many Singularities
I We show that the sequence (ζ(n))n≥2 of values of theRiemann zeta function is not holonomic.
I
feven(z) :=∑n≥1
ζ(2n)zn, fodd(z) :=∑n≥1
ζ(2n + 1)zn
I Euler:feven(z) = 1
2 −π2 z cotπz
I Molteni (2001):
feven(z)/z − fodd(z) = γ − ψ(1 + z),
where ψ(z) = Γ′(z)/Γ(z) is the digamma function.
Known Non-Trivial Results
I Harris, Sibuya (1985): The reciprocal 1/f of a holonomicfunction is holonomic iff f ′/f is algebraic.
I Singer (1986): Similar results about exponentiation andcomposition
I Van der Put, Singer (1997): The reciprocal 1/an of aholonomic sequence is holonomic iff an is an interlacement ofhypergeometric sequences.
I Polya-Carlson (1921): If f (z) has integer coefficients andradius of convergence 1, then f (z) is rational or has the unitcircle for its natural boundary.
Known Non-Trivial Results
I Harris, Sibuya (1985): The reciprocal 1/f of a holonomicfunction is holonomic iff f ′/f is algebraic.
I Singer (1986): Similar results about exponentiation andcomposition
I Van der Put, Singer (1997): The reciprocal 1/an of aholonomic sequence is holonomic iff an is an interlacement ofhypergeometric sequences.
I Polya-Carlson (1921): If f (z) has integer coefficients andradius of convergence 1, then f (z) is rational or has the unitcircle for its natural boundary.
Known Non-Trivial Results
I Harris, Sibuya (1985): The reciprocal 1/f of a holonomicfunction is holonomic iff f ′/f is algebraic.
I Singer (1986): Similar results about exponentiation andcomposition
I Van der Put, Singer (1997): The reciprocal 1/an of aholonomic sequence is holonomic iff an is an interlacement ofhypergeometric sequences.
I Polya-Carlson (1921): If f (z) has integer coefficients andradius of convergence 1, then f (z) is rational or has the unitcircle for its natural boundary.
Known Non-Trivial Results
I Harris, Sibuya (1985): The reciprocal 1/f of a holonomicfunction is holonomic iff f ′/f is algebraic.
I Singer (1986): Similar results about exponentiation andcomposition
I Van der Put, Singer (1997): The reciprocal 1/an of aholonomic sequence is holonomic iff an is an interlacement ofhypergeometric sequences.
I Polya-Carlson (1921): If f (z) has integer coefficients andradius of convergence 1, then f (z) is rational or has the unitcircle for its natural boundary.
Proofs by Number Theory (SG, 2004)
I W.l.o.g. the polynomials pk(n) have coefficients inQ(aj : j ≥ 0).
I an =√
n does not satisfy a recurrence
p0(n)an + · · ·+ pd(n)an+d = 0
with coefficients in Q(√
j : j ≥ 0)[n], since
[Q(√ρ1, . . . ,
√ρs) : Q] = 2s
for distinct primes ρ1, . . . , ρs .
I Other argument: transcendence of e implies non-holonomicityof nn.
Proofs by Number Theory (SG, 2004)
I W.l.o.g. the polynomials pk(n) have coefficients inQ(aj : j ≥ 0).
I an =√
n does not satisfy a recurrence
p0(n)an + · · ·+ pd(n)an+d = 0
with coefficients in Q(√
j : j ≥ 0)[n], since
[Q(√ρ1, . . . ,
√ρs) : Q] = 2s
for distinct primes ρ1, . . . , ρs .
I Other argument: transcendence of e implies non-holonomicityof nn.
Proofs by Number Theory (SG, 2004)
I W.l.o.g. the polynomials pk(n) have coefficients inQ(aj : j ≥ 0).
I an =√
n does not satisfy a recurrence
p0(n)an + · · ·+ pd(n)an+d = 0
with coefficients in Q(√
j : j ≥ 0)[n], since
[Q(√ρ1, . . . ,
√ρs) : Q] = 2s
for distinct primes ρ1, . . . , ρs .
I Other argument: transcendence of e implies non-holonomicityof nn.
Proofs by Asymptotics (P. Flajolet, SG, B. Salvy, 2005)
I Fuchs-Frobenius theory: Asymptotic expansion of holonomicfunctions as |z | → ∞ must be linear combination of series ofthe form
eP(z1/r )zα∑j≥0
Qj(log z)z−js ,
where P and Qj are polynomials, the Qj have boundeddegree, r ∈ N, α ∈ C, 0 < s ∈ Q.
I Hence log log z , eez−1 , and Lambert W are not holonomic.
Proofs by Asymptotics (P. Flajolet, SG, B. Salvy, 2005)
I Fuchs-Frobenius theory: Asymptotic expansion of holonomicfunctions as |z | → ∞ must be linear combination of series ofthe form
eP(z1/r )zα∑j≥0
Qj(log z)z−js ,
where P and Qj are polynomials, the Qj have boundeddegree, r ∈ N, α ∈ C, 0 < s ∈ Q.
I Hence log log z , eez−1 , and Lambert W are not holonomic.
Proofs by Asymptotics
I Basic Abelian theorem. Let φ(x) be any of the functions
xα(log x)β(log log x)γ , α ≥ 0, β, γ ∈ C. (1)
Let (an) be a sequence that satisfies the asymptotic estimate
an ∼n→∞
φ(n).
Then the generating function f (z) :=∑
n≥0 anzn satisfies
the asymptotic estimate
f (z) ∼z→1−
Γ(α+ 1)1
(1− z)φ
(1
1− z
). (2)
Proofs by Asymptotics
I The sequence of prime numbers:
n-th prime = n log n + n log log n + O(n),
hence(n-th prime)/n −Hn ∼ log log n.
I The sequences of powers (α ∈ C \ Z):
n∑k=1
(n
k
)(−1)kkα ∼ (log n)−α
Γ(1− α).
Proofs by Asymptotics
I The sequence of prime numbers:
n-th prime = n log n + n log log n + O(n),
hence(n-th prime)/n −Hn ∼ log log n.
I The sequences of powers (α ∈ C \ Z):
n∑k=1
(n
k
)(−1)kkα ∼ (log n)−α
Γ(1− α).
e1/n by Asymptotics (P. Flajolet, SG, B. Salvy, 2005)
I Lindelof integral representation
∑n≥1
e1/n(−z)n = − 1
2i
∫ 1/2+i∞
1/2−i∞
zse1/s
sinπsds
I Asymptotics (saddle point method)
∑n≥1
e1/n(−z)n ∼ − e2√
log z
2√π(log z)1/4
as |z | → ∞.
I Hence e1/n is not holonomic.
I Work in progress: generalize asymptotics to αnβ.
e1/n by Asymptotics (P. Flajolet, SG, B. Salvy, 2005)
I Lindelof integral representation
∑n≥1
e1/n(−z)n = − 1
2i
∫ 1/2+i∞
1/2−i∞
zse1/s
sinπsds
I Asymptotics (saddle point method)
∑n≥1
e1/n(−z)n ∼ − e2√
log z
2√π(log z)1/4
as |z | → ∞.
I Hence e1/n is not holonomic.
I Work in progress: generalize asymptotics to αnβ.
e1/n by Asymptotics (P. Flajolet, SG, B. Salvy, 2005)
I Lindelof integral representation
∑n≥1
e1/n(−z)n = − 1
2i
∫ 1/2+i∞
1/2−i∞
zse1/s
sinπsds
I Asymptotics (saddle point method)
∑n≥1
e1/n(−z)n ∼ − e2√
log z
2√π(log z)1/4
as |z | → ∞.
I Hence e1/n is not holonomic.
I Work in progress: generalize asymptotics to αnβ.
e1/n by Asymptotics (P. Flajolet, SG, B. Salvy, 2005)
I Lindelof integral representation
∑n≥1
e1/n(−z)n = − 1
2i
∫ 1/2+i∞
1/2−i∞
zse1/s
sinπsds
I Asymptotics (saddle point method)
∑n≥1
e1/n(−z)n ∼ − e2√
log z
2√π(log z)1/4
as |z | → ∞.
I Hence e1/n is not holonomic.
I Work in progress: generalize asymptotics to αnβ.
Closed-Form Sequences (J.P. Bell, SG, M. Klazar, F. Luca,2006)
I For sequences like nα, e1/n, there is a more direct way toexploit the additional structure
I an = f (n) for a smooth function f
I The recurrence means that
F (z ; p0, . . . , pd) :=d∑
k=0
pk(z)f (z + k)
vanishes at z = 0, 1, 2, . . . .
I Example: f (z) = zα. If F vanishes identically, the left handside of
f (z) = − 1
p0(z)
d∑k=1
pk(z)f (z + k)
is meromorphic at z = 0, hence α ∈ Z.
Closed-Form Sequences (J.P. Bell, SG, M. Klazar, F. Luca,2006)
I For sequences like nα, e1/n, there is a more direct way toexploit the additional structure
I an = f (n) for a smooth function f
I The recurrence means that
F (z ; p0, . . . , pd) :=d∑
k=0
pk(z)f (z + k)
vanishes at z = 0, 1, 2, . . . .
I Example: f (z) = zα. If F vanishes identically, the left handside of
f (z) = − 1
p0(z)
d∑k=1
pk(z)f (z + k)
is meromorphic at z = 0, hence α ∈ Z.
Closed-Form Sequences (J.P. Bell, SG, M. Klazar, F. Luca,2006)
I For sequences like nα, e1/n, there is a more direct way toexploit the additional structure
I an = f (n) for a smooth function f
I The recurrence means that
F (z ; p0, . . . , pd) :=d∑
k=0
pk(z)f (z + k)
vanishes at z = 0, 1, 2, . . . .
I Example: f (z) = zα. If F vanishes identically, the left handside of
f (z) = − 1
p0(z)
d∑k=1
pk(z)f (z + k)
is meromorphic at z = 0, hence α ∈ Z.
Zeroes of Analytic and of Elementary Functions
I Carlson’s theorem (1921). The function sinπz is the“smallest” function analytic for <(z) ≥ 0 that vanishes atz = 0, 1, 2, . . . .
I Shows non-holonomicity of nα, log n, e1/n, . . .
I Khovanskiı investigates the geometry of the zero set ofelementary functions in his book “Fewnomials”.
I Definition. Elementary functions are built by composingrational functions, exp(x), log(x), sin(x), cos(x), tan x ,arcsin(x), arccos(x), and arctan(x). The domain of definitionmust be such that arguments of sin and cos are bounded.
I Theorem (Khovanskiı). An elementary function has onlyfinitely many simple zeros in its domain of definition.
Zeroes of Analytic and of Elementary Functions
I Carlson’s theorem (1921). The function sinπz is the“smallest” function analytic for <(z) ≥ 0 that vanishes atz = 0, 1, 2, . . . .
I Shows non-holonomicity of nα, log n, e1/n, . . .
I Khovanskiı investigates the geometry of the zero set ofelementary functions in his book “Fewnomials”.
I Definition. Elementary functions are built by composingrational functions, exp(x), log(x), sin(x), cos(x), tan x ,arcsin(x), arccos(x), and arctan(x). The domain of definitionmust be such that arguments of sin and cos are bounded.
I Theorem (Khovanskiı). An elementary function has onlyfinitely many simple zeros in its domain of definition.
Zeroes of Analytic and of Elementary Functions
I Carlson’s theorem (1921). The function sinπz is the“smallest” function analytic for <(z) ≥ 0 that vanishes atz = 0, 1, 2, . . . .
I Shows non-holonomicity of nα, log n, e1/n, . . .
I Khovanskiı investigates the geometry of the zero set ofelementary functions in his book “Fewnomials”.
I Definition. Elementary functions are built by composingrational functions, exp(x), log(x), sin(x), cos(x), tan x ,arcsin(x), arccos(x), and arctan(x). The domain of definitionmust be such that arguments of sin and cos are bounded.
I Theorem (Khovanskiı). An elementary function has onlyfinitely many simple zeros in its domain of definition.
Zeroes of Analytic and of Elementary Functions
I Carlson’s theorem (1921). The function sinπz is the“smallest” function analytic for <(z) ≥ 0 that vanishes atz = 0, 1, 2, . . . .
I Shows non-holonomicity of nα, log n, e1/n, . . .
I Khovanskiı investigates the geometry of the zero set ofelementary functions in his book “Fewnomials”.
I Definition. Elementary functions are built by composingrational functions, exp(x), log(x), sin(x), cos(x), tan x ,arcsin(x), arccos(x), and arctan(x). The domain of definitionmust be such that arguments of sin and cos are bounded.
I Theorem (Khovanskiı). An elementary function has onlyfinitely many simple zeros in its domain of definition.
Results proved using Carlson or Khovanskiı
I For distinct complex u1, . . . , us , the sequenceΓ(n − u1)
α1 . . . Γ(n − us)αs is holonomic if and only if
α1, . . . , αs are integers.
I If a sequence from R(n, en) is holonomic, then thedenominator has just one summand.
I If (f (n))n≥1 is holonomic for an algebraic functionf : ]1,∞] → R, then f is a rational function.
Results proved using Carlson or Khovanskiı
I For distinct complex u1, . . . , us , the sequenceΓ(n − u1)
α1 . . . Γ(n − us)αs is holonomic if and only if
α1, . . . , αs are integers.
I If a sequence from R(n, en) is holonomic, then thedenominator has just one summand.
I If (f (n))n≥1 is holonomic for an algebraic functionf : ]1,∞] → R, then f is a rational function.
Results proved using Carlson or Khovanskiı
I For distinct complex u1, . . . , us , the sequenceΓ(n − u1)
α1 . . . Γ(n − us)αs is holonomic if and only if
α1, . . . , αs are integers.
I If a sequence from R(n, en) is holonomic, then thedenominator has just one summand.
I If (f (n))n≥1 is holonomic for an algebraic functionf : ]1,∞] → R, then f is a rational function.
Conclusion
I “Negative” results, but “positive” ones obtained along the way
I Opportunity to apply methods and results from various areas
I There is a good chance that holonomicity of a sequence canbe decided if it has (i) a closed form representation or (ii) aknown asymptotic expansion.
Conclusion
I “Negative” results, but “positive” ones obtained along the way
I Opportunity to apply methods and results from various areas
I There is a good chance that holonomicity of a sequence canbe decided if it has (i) a closed form representation or (ii) aknown asymptotic expansion.
Conclusion
I “Negative” results, but “positive” ones obtained along the way
I Opportunity to apply methods and results from various areas
I There is a good chance that holonomicity of a sequence canbe decided if it has (i) a closed form representation or (ii) aknown asymptotic expansion.