Padé approximations of generalized …cc.oulu.fi/~tma/TOKYOSLIDES.pdfAbstract We shall present short proofs for type II Pad e approximations of the generalized hypergeometric and

Pade approximations of generalized

hypergeometric series

Tapani Matala-aho

Tokyo 2009 March 02

Abstract

We shall present short proofs for type II Pade approximations of

the generalized hypergeometric and q-hypergeometric series

F (t) =∞∑

n=0

∏n−1k=0 P(k)∏n−1k=0 Q(k)

tn, Fq(t) =∞∑

n=0

∏n−1k=0 P(qk)∏n−1k=0 Q(qk)

tn. (1)

In a q-exponential case we will discuss how certain modified

approximations give sharp linear independence results. Further, a

comparison is done between the remainder series approximations of

the exponential series (Prevost and Rivoal) and our modified

approximations for a q-analogue of the exponential series.

Arithmetic Motivation

An interesting part of Number Theory is involved with a question

of arithmetic nature of explicitly defined numbers.

-Irrationality

-Linear independence over a field

-Transcendency

Even more interesting with a quantitative setting.

Generalized Hypergeometric series

Let P(y) and Q(y) be polynomials and define generalized

(classical) hypergeometric and (basic) q-hypergeometric series

F (t) =∞∑

n=0

[P]n[Q]n

tn, Fq(t) =∞∑

n=0

[P; q]n[Q; q]n

tn, (2)

where

[P]n = [P(y)]n =n−1∏k=0

P(k), (3)

[P; q]n = [P(y); q]n =n−1∏k=0

P(qk). (4)

Classical hypergeometric series

Pochhammer symbol (generalized factorial)

(a)0 = 1, (a)n = a(a + 1) · · · (a + n − 1) n ∈ Z+. (5)

(1)n = n!

Hypergeometric series

AFB

(a1, ..., aA

b1, ..., bB

∣∣∣ t) =∞∑

n=0

(a1)n · · · (aA)n

n!(b1)n · · · (bB)ntn (6)

Classical case

First we will study the classical series F (t) with it’s derivativies

∆bF (t), where ∆ = t ddt .

Denote d = max{deg P(y), deg Q(y)} and let d ,m ∈ Z+ and the

numbers α1, ..., αm be given.

We start by giving explicit type II Pade approximations for the

series

∆bF (tαj), b = 0, 1, ..., d − 1; j = 1, ...,m. (7)

Our construction is based on a product expansion a la Maier

[Potenzreihen irrationalen Grenzwertes. J. Reine Angew. Math.

156, 93–148 (1927)]

Maier’s product formula

Let l ,m ∈ Z+ and α = t(α1, ..., αm) be given and define

σi = σi (l , α) bym∏

t=1

(αt − w)l =ml∑i=0

σiwi . (8)

Thenml∑i=0

σi ikαi

t = 0 (9)

for all t ∈ {1, ...,m}; k ∈ {0, ..., l − 1}.

Maier’s product formula

Moreover

σi = (−1)i∑

i1+...+im=i

(l

i1

)· · ·(

l

im

)· αl−i1

1 · · ·αl−imm . (10)

Denote

Σh =∑

h1+...+hm=h

(l

h1

)· · ·(

l

hm

)· αh1

1 · · ·αhmm (11)

Pade approximations/Classical case

Let b, d , l ,m, λ ∈ N, b < d and choose m numbers α1, ..., αm. Put

Bl ,λ(t) =ml∑i=0

tml−iσi (l , α)[Q]i+λ+bl/dc−1

[P]i+λ. (12)

Then

Bl ,λ(t)∆bF (αj t)− Al ,λ,b,j(t) = Rl ,λ,b,j(t), (13)

where

degt Bl ,λ(t) = ml , degt Al ,λ,b,j(t) ≤ ml + λ− 1 (14)

ordt=0

Rl ,λ,b,j(t) ≥ ml + bl/dc+ λ. (15)

Pade approximations/Classical case

Thus we have a gap of lenght bl/dc in the power series expansion

Bl ,λ(t)∆bF (αj t) = Al ,λ,b,j(t) + Rl ,λ,b,j(t). (16)

The polynomials Bl ,λ(t) are Pade approximant denominators in

variable t for the functions Fb,j(t) = ∆bF (tαj),

b = 0, 1, ..., d − 1; j = 1, ...,m.

Also we say that (13–15) define a Pade approximation with the

degree and order parameters

[degt B, degt A ≤, ordt=0

R ≥] = [ml ,ml + λ− 1,ml + bl/dc+ λ]

(17)

Proof

Write

Bl ,λ(t) =ml∑

h=0

thbl ,λ,h =ml∑i=0

tml−iσi

[Q]i+λ+bl/dc−1

[P]i+λ(18)

and study the expansion of the product

Bl ,λ(t)∆bF (tαj) =∞∑

N=0

rNtN , (19)

where

rN =∑

h+n=N

bl ,λ,hnbfnα

nj (20)

Proof

Put

N = ml + λ+ a (21)

Then

rN =ml∑i=0

σi

[Q]i+λ+bl/dc−1

[P]i+λ

[P]i+λ+a

[Q]i+λ+a(i + λ+ a)bαi+λ+a

j (22)

or

Proof

rN =ml∑i=0

σi[P]i+λ+a

[P]i+λ

[Q]i+λ+bl/dc−1

[Q]i+λ+a(i + λ+ a)bαi+λ+a

j (23)

If now

0 ≤ a ≤ bl/dc − 1 (24)

then

rN = αa+λj

ml∑i=0

σi (i + λ+ a)bαij ·

[P(i + λ)]a[Q(i + λ+ a)]bl/dc−a−1 (25)

Proof

By binomial theorem we may denote

(i + a + λ)b =b∑

k=0

sk ik , (26)

P(i + b) =d∑

kb=0

pkbik , Q(i + b) =

d∑kb=0

qkbik ∀b ∈ Z,

where pb, qb, sk ’s are independent of i .

Proof

Thus

[P(i + λ)]a =

ad∑K=0

∑k0+...+ka−1=K

pk0 · · · pka−1 ik0 · · · ika−1

=ad∑

K=0

PK iK , (27)

and similarly

[Q(qa+i+λ)]bl/dc−a−1 =

(bl/dc−a−1)d∑K=0

QK iK , (28)

where PK ,QK ’s are independent of i .

Proof

So,

rN = αa+λj

(a+bl/dc−a−1)d+b∑K=0

∑k+K1+K2=K

skPK1QK2

ml∑i=0

σiαij i

k+K1+K2

(29)

and finally

rN = αa+λj

(bl/dc−1)d+b∑K=0

∑k+K1+K2=K

skPK1QK2

ml∑i=0

σi iKαi

j , (30)

where 0 ≤ K ≤ (bl/dc − 1)d + b ≤ l − 1.

Proof

Hence, by using the property

ml∑i=0

σi ikαi

j = 0 (31)

for j ∈ {1, ...,m}; k ∈ {0, ..., l − 1} we have

rN = rml+λ+a = 0 ∀ 0 ≤ a ≤ bl/dc − 1. (32)

NOTE: The crucial point in the proof is formula (23) which in fact

shows how to find the right denominator polynomials.

q-world

q-series factorials (q-Pochhammer symbols):

(b, a)0 = (b, a; q)0 = 1

(b, a)n = (b, a; q)n = (b − a)(b − aq)...(b − aqn−1), n ∈ Z+

(a)n = (a; q)n = (1, a)n = (1− a)(1− aq) · · · (1− aqn−1) (33)

(q)n = (q; q)n = (1− q)...(1− qn)

q-hypergeometric (basic) series

AΦB

(a1, ..., aA

b1, ..., bB

∣∣∣ t) =∞∑

n=0

(a1)n...(aA)n

(q)n(b1)n...(bB)ntn. (34)

q-world

The q-binomial coefficients are defined by[nk

]=

(q; q)n

(q; q)k(q; q)n−k(35)

q-binomial theorem [2, p. 490, Corollary 10.2.2(c)]:

(b, a; q)n =n∑

k=0

[nk

]q(k

2)bn−k(−a)k ∀n ∈ N (36)

Pade approximations/q-world

Stihl [Arithmetische Eigenschaften spezieller Heinescher Reihen.

Math. Ann. 268, 21–41 (1984)]

Let l ,m ∈ Z+ and α = t(α1, ..., αm) be given and define

σq,i = σq,i (l , α) by

m∏t=1

(αt ,w ; q−1)l =ml∑i=0

σq,iwi (37)

Thenml∑i=0

σq,i (αjqk)i = 0 (38)

for all j ∈ {1, ...,m}; k ∈ {0, ..., l − 1}.


Moreover, by q-binomial theorem,

σq,i = (−1)iq−m( l2)Σq,ml−i (39)

holds with

Σq,h = Σq,h(l , α) =∑

i1+...+im=h

[l

i1

]· · ·[

l

im

]·

q(i12)+...+(im

2 )αi11 · · ·α

imm . (40)


Define a operator J = Jt by Jf (t) = f (qt) and denote again

d = max{deg P(y), deg Q(y)}.

Stihl constructed the following explicit type II Pade approximations

in variable t for the d series JbF (t), 0 ≤ b ≤ d − 1 at m points.


Let b, l ,m, λ ∈ N, 0 ≤ b < d = max{deg P(y), deg Q(y)} and

choose m numbers α1, ..., αm. Put σq,i = σq,i (l , α) and

Bl ,λ(t) =ml∑

h=0

thbl ,λ,h =ml∑i=0

tml−iσq,i

[Q; q]i+λ+bl/dc−1

[P; q]i+λ(41)

Then

Bl ,λ(t)JbF (αj t)− Al ,λ,b,j(t) = Rl ,λ,b,j(t), (42)

holds with the parameters

[ml ,ml + λ− 1,ml + bl/dc+ λ]. (43)


This means that the polynomials Bl ,λ(t) are Pade approximant

denominators in variable t for the functions Fb,j(t) = JbF (tαj),

b = 0, 1, ..., d − 1; j = 1, ...,m.

The linear independence results by applying the above explicit

Pade approximations of q-hypergeometric series restrict usually to

the following class of functions

F (t) =∞∑

n=0

qm(n2)

(q)n(a1)n...(al)ntn, m ≥ 2, l ≤ m − 2 (44)

with |q| < 1.

q-world/analogues of exponential series

Hence, for example, the linear independence question of

q-analogues of exponential function and Euler’s divergent series

have resisted the attacts from the explicit Pade approximations.

q-analogues of exponential series:

Eq(z) =∞∑

n=0

1

(q; q)nzn, Eq(z) =

∞∑n=0

q(n2)

(q; q)nzn (45)

A q-analogue of Euler’s divergent series:

Dq(t) =∞∑

n=0

(q; q)ntn. (46)

Arithmetic of q-series

Amou M., Andre Y., Bertrand D., Bezivin, Borwein P., Bundschuh

P., Duverney D., Katsurada M., Nesterenko Yu., Nishioka K.,

Prevost M., Rivoal T., Stihl Th., Shiokawa I., Waldscmidt M.,

Wallisser R., Vaananen K., Zudilin W.

q-world numbers

p-adic, p ∈ P:∞∑

n=1

pn

1− pn, (47)

∞∑n=1

pn

n∏i=1

1± pi

, (48)

∞∑l=0

pl2∏lj=1(1± pj)2

(49)

q-world numbers

1 +p

1 +

p2

1 +

p3

1 + . . .(50)

∞∏n=1

(1 + kpn), k = 1, ..., p − 1, (51)

∞∑n=1

pnn∏

i=1

(1 + kpi ), k = 1, ..., p − 1, (52)

q-world numbers

Real, p ∈ Z \ {0,±1}:∞∑

n=1

1

1− pn, (53)

∞∑n=1

1n∏

i=11± pi

, (54)

∞∑l=0

1∏lj=1(1± pj)2

, (55)

q-world numbers

1 +p−1

1 +

p−2

1 +

p−3

1 + . . ., (56)

∞∏n=1

(1 + kp−n), k = 0, 1, ..., p − 1, (57)

∞∑n=1

p−nn∏

i=1

(1 + kp−i ), k = 0, 1, ..., p − 1. (58)

q-world numbers

1

F1 +

1

F3 +

1

F5 + . . ., (59)

1

F2 +

1

F4 +

1

F6 + . . ., (60)

∞∑n=0

1

Fan+b, (61)

∞∑n=0

1

Lan+b, (62)

where a, b,∈ Z+, Fn and Ln are the Fibonacci and Lucas numbers,

respectively; F0 = 0,F1 = 1, L0 = 2, L1 = 1.

q-world/An analogue of Euler’s divergent series

Euler’s divergent series:∞∑

n=0

n!tn. (63)

A q-analogue of Euler’s divergent series:

Dq(t) =∞∑

n=0

(q; q)ntn. (64)

a special case of

Db(t) =∞∑

n=0

(b; q)ntn (65)

Modified Maier-Stihl/q-world

In the following we will present explicit Pade type approximations

in variable t for the q-exponential series

Fx(t) =∞∑

k=0

q(k2)(−tx)k

(t; q)k+1. (66)

First, we have

Fx(t) = Dx(t) =∞∑

n=0

(x ; q)ntn (67)


Define q-factorial polynomials

(x ; q)n =n−1∏h=0

(1− xqh) =n∑

k=0

s(n, k)xk (68)

and set s(n, k) = 0, when k < 0 or n < k .

Thenn∑

k=0

s(n, k)(−x)n−k = q(n2)xn ∀ n ∈ N (69)

which plays an essential role in the following modification.


Define the following polynomials

Bl ,ν(t) =ml∑

h=0

bl ,ν,h(t)th, (70)

with

bl ,ν,h(t) = (−1)ml−hq(ml+ν2 )−(ml+ν−h

2 )(t; q)ml+ν−hΣq,h (71)


Then by (68)

Bl ,ν(t) =ml+ν∑H=0

bl ,ν,HtH (72)

with

bl ,ν,H = qm( l2)

∑ml−i+f =H

0≤f≤i+ν≤ml+ν

q(ml+ν2 )−(i+ν

2 )s(i + ν, f )σq,i (73)

Modified Pade approximations

Now we get the following modified Pade type approximations of

the second kind for the m functions D−αj (t), (j = 1, ...,m) in

variable t. Let l , ν ∈ N and j = 1, ...,m, then

Bl ,ν(t)D−αj (t) + Al ,ν,j(t) = Ll ,ν,j(t) (74)

with

[ml + ν,ml + ν − 1, (m + 1)l + ν] (75)

By (75) we have a diagonal type Pade approximation with the free

parameter ν.


Further

Al ,ν,j(t) =ml+ν−1∑

N=0

al ,ν,j ,NtN , (76)

where

al ,ν,j ,N = −∑

H+n=N

bl ,ν,H(−αj ; q)n, (77)

and


Ll ,ν,j(t) = t(m+1)l+νq(ml+ν2 )+m( l

2)+νl(−αj ; q)lανj Sl ,ν,j(t). (78)

where

Sl ,ν,j(t) =∞∑

k=0

sl ,ν,j ,ktk (79)

with

sl ,ν,j ,k = qkν(−αjql ; q)k

m∏t=1

(αt , αjqk+1; q)l . (80)

Modified Maier-Stihl/Proof

The expansion

Bl ,ν(t)D−αj (t) =∞∑

N=0

rNtN , (81)

holds with

rN =∑

H+n=N

bl ,ν,H(−αj ; q)n. (82)


Set

N = ml + ν + a 0 ≤ a ≤ l − 1

then

rN = qm( l2)

ml∑i=0

i+ν∑f =0

q(ml+ν2 )−(i+ν

2 )s(i + ν, f )σq,i (−αj ; q)i+ν−f +a =

qm( l2)

ml∑i=0

σq,i (−αj ; q)aq(ml+ν

2 )−(i+ν2 )·

i+ν∑f =0

s(i + ν, f )(−αjqa; q)i+ν−f (83)


Here the inner f -sum is evaluated by (69) and so

rml+ν+a = qm( l2)+(ml+ν

2 )(−αj ; q)a

ml∑i=0

σq,i (αjqa)i+ν =

qm( l2)+(ml+ν

2 )+aν(−αj ; q)aανj

m∏t=1

(αt , αjqa; q−1)l = 0 (84)

for any 0 ≤ a ≤ l − 1 .


Next we consider the case a = l + k , k ∈ N. Then

rN = r(m+1)l+ν+k =

qm( l2)+(ml+ν

2 )+(l+k)ν(−αj ; q)l+kανj

m∏t=1

(αt , αjqk+1; q)l . (85)

Modified Maier-Stihl

Note that by Stihl

C (x , t) =ml∑

h=0

(xt)h(t; q)(m+1)l+λ−h· (86)

(−1)ml−hq(ml+λ+12 )−(ml+λ−h+1

2 )Σq,h

is a denominator polynomial in variable x and

Bl ,ν(t) =ml∑

h=0

(tx)h(t; q)ml+ν−h (87)

(−1)ml−hq(ml+ν2 )−(ml+ν−h

2 )Σq,h

is a denominator polynomial in variable t for

Fx(t) =∞∑

k=0

q(k2)(−tx)k

(t; q)k+1. (88)


By using the above modified approximations we may prove linear

independence measures for certain values of the functions Da(z)

and Ea(z), which can be regarded as q-analogues of Euler’s

divergent series and the usual exponential series.

For the q-exponential function Eq(z), our result asserts the linear

independence (over any number field) of the values 1 and Eq(αj)

(j = 1, ...,m) together with its measure having the exponent

ω = O(m), which sharpens the known exponent ω = O(m2)

obtained by a certain refined version of Siegel’s lemma Amou,

Matala-aho, Vaananen [On Siegel-Shidlovskii’s theory for

q-difference equations. Acta Arith. 127, 309–335 (2007)]


Let p be a prime number. Then we have the linear independence

of the p-adic numbers

∞∏n=1

(1 + kpn), k = 0, 1, ..., p − 1 (89)

over Q with a measure having the exponent

ω < 2p (90)

Valuations, heights

Let K be a fixed number field of degree κ = [K : Q], v a place of

K and | |v the associated absolute value on the completion Kv

with a local degree κv = [Kv : Qv ].

If the finite place v of K lies over the prime p, we write v |p, for an

infinite place v of K we write v |∞. We normalize the absolute

value | |v of K so that

|p|v = p−1, if v |p, (91)

|x |v = |x |, if v |∞, (92)

where | | denotes the ordinary absolute value in Q.

Valuations, heights

Further, the notation

||α||v = |α|κv/κv , κv = [Kv : Qv ], (93)

will be used in the sequel. The height H(α) of α is defined by the

formula

H(α) =∏v

||α||∗v , ||α||∗v = max{1, ||α||v} (94)

and the height H(α) of the vector α = t(α1, ..., αm) ∈ Km is given

by

H(α) =∏v

||α||∗v , ||α||∗v = maxi=1,...,m

{1, ||αi ||v}. (95)

Valuations, heights

Further, for any place v of K, and q ∈ K∗, ||q||v 6= 1, we define

the characteristic λ by

λ = λq =log H(q)

log ||q||v. (96)

Modified Maier-Stihl/Applications

To state our results we denote

ω = ωq =u0

u0 + λqs0, (97)

where

s0 = m2 + m + m√

m2 + m, (98)

u0 = m2 + m + (m + 1)√

m2 + m. (99)

Now we fix a place v of K throughout the following.


Let m ∈ Z+ be arbitrary, a, q, α1, ..., αm ∈ K∗, and |q|v < 1.

Denote by f (t) each of the functions

Dt(a), Eq(t),∞∏

n=0

(1− tqn) (100)

and assume

a /∈ q−N, αi /∈ q−N, αi /∈ αjqZ for all i 6= j , (101)

−(

1 +1

m +√

m2 + m

)< λq ≤ −1. (102)


Then the numbers 1, f (α1), ..., f (αm) belonging to Kv are linearly

independent over K. Further, there exist positive constants

c , d ,H0 depending on a and αi such that

|k0 + k1f (α1) + ...+ kmf (αm)|v >c

Hωκ/κv+d(log H)−1/2(103)

for all

k = t(k0, k1, ..., km) ∈ Km+1 \ {0}

with H = max(H(k),H0).


Here we note that λq ≤ −1 always holds for |q|v < 1, and the

following cases in particular assert λq = −1:

1. K = I, v is the infinite place of K, and 1/q ∈ ZK;

2. K = Q, v = p ∈ P, and q = pl , l ∈ Z+;

3. K = Q(√

5), q = ((1−√

5)/2)l , l ∈ Z+.

Now, if we take the value λ = −1, then we have

ω = m + 1 +√

m2 + m < 2m + 2 (104)

and in general we have ω = O(m),too.


In the cases 1–3 we have

|L0 + L1f (α1) + ...+ Lmf (αm)|p >c

H2m+2+d(log H)−1/2(105)

for all

L = t(L0, L1, ..., Lm) ∈ Zm+1 \ {0}

with H = maxi=0,...,m

(|Li |,H0).


Let p ∈ P. Then in each of the four sets∞∏

n=1

(1 + kpn), k = 0, 1, ..., p − 1, (106)

∞∑n=1

pnn∏

i=1

(1 + kpi ), k = 0, 1, ..., p − 1, (107)

∞∏n=1

(1 + kp−n), k = 0, 1, ..., p − 1, (108)

∞∑n=1

p−nn∏

i=1

(1 + kp−i ), k = 0, 1, ..., p − 1. (109)

of p numbers we have the linear independence of the p numbers

over Q with a measure having an exponent

ω < 2p. (110)

q-world

q-difference equations

Eq(qz) = (1− z)Eq(z), Eq(z) = (1 + z)Eq(qz). (111)

By using (111) one gets the well-known Euler formulae

Eq(z) =1

(z)∞, Eq(z) = (−z)∞. (112)

q-identities

From (112) we get

Eq(t)Eq(−t) = 1 (113)

which implies

E1/q,1/q(t)Eq,q(qt) = 1 (114)

The following connects q-exponentials and q-divergent

∞∑n=0

q(n2)αn

(qz)nzn = 1 + αzD−α,q(qz) (115)

Remaider series technique

Prevost and Rivoal [Remainder Pade Approximants for the

Exponential Function. Constr. Approx. 25, 109–123 (2007)]

presented new Pade type approximations for the classical

exponential series using the remainder series technique, invented in

Prevost [A new proof of the irrationality of ζ(2) and ζ(3) using

Pade approximants. J. Comput. Appl. Math. 67, 219–235 (1996)].

To be more precise, they constructed approximations in variable t

for the exponential remainder series

Φxαj (t) =∞∑

k=0

(xαj)k

(1− 1/t)k, j = 1, ...,m (116)

Remaider series technique/Modified Maier

It is interesting that the above series Φx(t) and Dx(t) share a

formal similarity. Namely

Φx(t) =∞∑

k=0

(−tx)k

[1− ty ]k+1=∞∑

k=0

xk

(1− 1/t)k; (117)

[1− ty ]k+1 = (1− t · 0)(1− t · 1)(1− t · 2) · · · (1− t · k) (118)

and

Dx(t) =∞∑

k=0

q(k2)(−tx)k

[1− ty ; q]k+1= −1

t

∞∑k=0

(x/q)k

(1/t; 1/q)k+1; (119)

[1− ty ; q]k+1 = (1− tq0)(1− tq1)(1− tq2) · · · (1− tqk) (120)


By applying the above modified Maier’s method we may give

another proof for the new Pade approximations for the exponential

remaider series

Φx(t) =∞∑

k=0

(−tx)k

[1− ty ]k+1=∞∑

k=0

xk

(1− 1/t)k. (121)

given by Prevost and Rivoal. Prevost and Rivoal showed that

Φx(t) =∞∑

n=0

Tn(−x)tn, (122)

where Tn(x) are Touchard polynomials


Tn(x) =n∑

k=0

S2(n, k)xk (123)

defined with Stirling numbers of the second kind S2(n, k). Further,

In the following s1(n, k) denote the Stirling numbers of first kind.

So our aim is to construct explicit simultaneous Pade type

approximations in variable t for the series

Φ−αj (t) =∞∑

n=0

Tn(αj)tn, j = 1, ...,m. (124)


We now define

Bl ,ν(t) =ml∑

h=0

bl ,ν,h(t)th, (125)

with

bl ,ν,h(t) = (−1)ml−h[1− ty ]ml+ν−hΣh(α), (126)

Then

Bl ,ν(t) =ml+ν∑H=0

bl ,ν,HtH , (127)

with

bl ,ν,H =∑

ml−i+f =H0≤f≤i+ν≤ml+ν

s1(i + ν, i + ν − f )σi (α) (128)


Let l , ν,m ∈ N and choose m numbers α1, ..., αm. Then

Bl ,ν(t)Φ−αj (t) + Al ,ν,j(t) = Ll ,ν,j(t) (129)

with

[ml + ν,ml + ν − 1, (m + 1)l + ν] (130)

give a diagonal type Pade approximation of the second kind with a

free parameter ν.

Remaider series technique/Modified Maier/Proof

Now we note that

[1− ay ]k = (1−a ·0)(1−a ·1)(1−a ·2) · · · (1−a ·(k−1)) = (131)

ak(1/a)k =k∑

i=0

s1(k , i)ak−i =k∑

i=0

s1(k , k − i)ai

Thus

Bl ,ν(t) =ml∑i=0

tml−i [1− ty ]i+νσi =

ml+ν∑H=0

tH∑

ml−i+f =H0≤f≤i+ν≤ml+ν

s1(i + ν, i + ν − f )σi =

ml+ν∑H=0

bl ,ν,HtH , (132)


We next study the expansion of the product

Bl ,ν(t)Φ−αj (t) =∞∑

N=0

rNtN , (133)

where

rN =∑

H+n=N

bl ,ν,HTn(αj) (134)

Set

N = ml + ν + a, 0 ≤ a ≤ l − 1, a ∈ N. (135)

Then

H = ml − i + f , H + n = N. (136)


It follows that n = i + ν − f + a and thus

rN =ml∑i=0

σi

i+ν∑f =0

s1(i + ν, i + ν − f )Ti+ν−f +a(αj) =

ml∑i=0

σi

i+ν∑f =0

a+ν+i−f∑e=0

s1(i + ν, i + ν − f )S2(a + ν + i − f , e)αej (137)

Denote for shortly I = i + ν, K = I − f . Then we are led to study

the following inner sums


Ha =I∑

K=0

a+K∑e=0

s1(I ,K )S2(a + K , e)αej =

a+I∑e=0

αej

I∑K=e−a

s1(I ,K )S2(a + K , e) =

a+I∑e=0

αej

I∑K=0

s1(I ,K )S2(a + K , e) (138)

Generalized Stirling orthogonality

Let a ∈ N. For all I , e ∈ N we have

I∑K=0

s1(I ,K )S2(a + K , e) =∑

0≤b,d≤a

Cb,d(a)edδI ,e−b (139)

where the numbers Cb,d(a) do not depend on I and e.


First

H0 =I∑

e=0

αej

I∑K=0

s1(I ,K )S2(K , e) =

I∑e=0

αej δIe = αI

j = αi+νj (140)

which by (9) shows

rml+ν =ml∑i=0

σiαi+νj = ανj

ml∑i=0

σiαij = 0. (141)


Further

H1 =I∑

e=0

αej

I∑K=0

s1(I ,K )S2(K + 1, e) =

I∑e=0

αej (δI ,e−1 + eδIe) = αI+1

j + IαIj = αi+ν+1

j + (i + ν)αi+νj (142)

which by (9) shows

rml+ν+1 =ml∑i=0

σiαi+ν+1j + (i + ν)αi+ν

j =

(αν+1j + νανj )

ml∑i=0

σiαij + ανj

ml∑i=0

σi iαij = 0. (143)


And in full generality by (139) we have

Ha =a+I∑e=0

αej

∑0≤b,d≤a

Cb,dedδI ,e−b =

∑0≤b,d≤a

Cb,d

a+I∑e=0

αej e

dδI ,e−b =

∑0≤b,d≤a

Cb,dαI+bj (I + b)d =

∑0≤b,d≤a

Cb,dαi+ν+bj (i + ν + b)d =

∑0≤b,d≤a

Cb,dαi+ν+bj

d∑g=o

(d

g

)ig (b + ν)d−g (144)


which gives

rml+ν+a =ml∑i=0

σiHa =

∑0≤b,d≤a

d∑g=0

Cb,dαi+ν+bj

(d

g

)(b + ν)d−g

ml∑i=0

σiαij i

g (145)

where 0 ≤ g ≤ a ≤ l − 1. Hence, again by using (9) we may

deduce

rml+ν+a = 0 (146)

for any 0 ≤ a ≤ l − 1 .

References

I [1]. Amou M., Matala-aho T., Vaananen K.: On

Siegel-Shidlovskii’s theory for q-difference equations. Acta

Arith. 127, 309–335 (2007)

I [2]. Andrews G., Askey R., Roy R.: Special Functions,

Encyclopedia of Mathematics and its Applications, 71.

Cambridge University Press, Cambridge, 1999.

I [3]. Chudnovsky G. V.: Pade approximations to the

generalized hypergeometric functions. I. J. Math. pures et

appl. 58, 445–476 (1979)

I [4]. Hata M.: Remarks on Mahler’s Transcendence Measure

for e. J. Number Theory 54, 81–92 (1995)

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Padé approximations of generalized …cc.oulu.fi/~tma/TOKYOSLIDES.pdfAbstract We shall present short proofs for type II Pad e approximations of the generalized hypergeometric and