Padé approximations of generalized hypergeometric series

Transcription

1 Padé approximations of generalized hypergeometric series Tapani Matala-aho Tokyo 2009 March 02

2 Abstract We shall present short proofs for type II Padé approximations of the generalized hypergeometric and q-hypergeometric series F (t) = n=0 n 1 k=0 P(k) n 1 k=0 Q(k)tn, F q (t) = n=0 n 1 k=0 P(qk ) n 1 k=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 (Prévost and Rivoal) and our modified approximations for a q-analogue of the exponential series.

3 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.

4 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 t n, F q (t) = n=0 [P; q] n [Q; q] n t n, (2) where [P] n = [P(y)] n = [P; q] n = [P(y); q] n = n 1 k=0 n 1 k=0 P(k), (3) P(q k ). (4)

5 Classical hypergeometric series Pochhammer symbol (generalized factorial) (a) 0 = 1, (a) n = a(a + 1) (a + n 1) n Z +. (5) Hypergeometric series (1) n = n! ( a1,..., a ) A AF B t = b 1,..., b B n=0 (a 1 ) n (a A ) n n!(b 1 ) n (b B ) n t n (6)

6 Classical case First we will study the classical series F (t) with it s derivativies b F (t), where = t d dt. Denote d = max{deg P(y), deg Q(y)} and let d, m Z + and the numbers α 1,..., α m be given. We start by giving explicit type II Padé approximations for the series b F (tα j ), 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, (1927)]

7 Maier s product formula Let l, m Z + and α = t (α 1,..., α m ) be given and define σ i = σ i (l, α) by Then m ml (α t w) l = σ i w i. (8) t=1 i=0 ml i=0 for all t {1,..., m}; k {0,..., l 1}. σ i i k α i t = 0 (9)

8 Maier s product formula Moreover σ i = ( 1) i i i m=i ( l i 1 ) ( ) l α l i 1 1 αm l im. (10) i m Denote Σ h = h h m=h ( l h 1 ) ( ) l α h 1 1 h αhm m (11) m

9 Padé approximations/classical case Let b, d, l, m, λ N, b < d and choose m numbers α 1,..., α m. Put ml B l,λ (t) = t ml i σ i (l, α) [Q] i+λ+ l/d 1. (12) [P] i+λ i=0 Then B l,λ (t) b F (α j t) A l,λ,b,j (t) = R l,λ,b,j (t), (13) where deg t B l,λ (t) = ml, deg t A l,λ,b,j (t) ml + λ 1 (14) ord R l,λ,b,j(t) ml + l/d + λ. (15) t=0

10 Padé approximations/classical case Thus we have a gap of lenght l/d in the power series expansion B l,λ (t) b F (α j t) = A l,λ,b,j (t) + R l,λ,b,j (t). (16) The polynomials B l,λ (t) are Padé approximant denominators in variable t for the functions F b,j (t) = b F (tα j ), b = 0, 1,..., d 1; j = 1,..., m. Also we say that (13 15) define a Padé approximation with the degree and order parameters [deg t B, deg t A, ord t=0 R ] = [ml, ml + λ 1, ml + l/d + λ] (17)

11 Proof Write ml ml B l,λ (t) = t h b l,λ,h = t ml i [Q] i+λ+ l/d 1 σ i (18) [P] i+λ h=0 i=0 and study the expansion of the product B l,λ (t) b F (tα j ) = r N t N, (19) N=0 where r N = b l,λ,h n b f n αj n (20) h+n=n

12 Proof Put N = ml + λ + a (21) Then or ml r N = i=0 σ i [Q] i+λ+ l/d 1 [P] i+λ [P] i+λ+a [Q] i+λ+a (i + λ + a) b α i+λ+a j (22)

13 Proof ml [P] i+λ+a r N = σ i [P] i+λ i=0 [Q] i+λ+ l/d 1 (i + λ + a) b α i+λ+a j (23) [Q] i+λ+a If now then r N = α a+λ j 0 a l/d 1 (24) ml i=0 σ i (i + λ + a) b α i j [P(i + λ)] a [Q(i + λ + a)] l/d a 1 (25)

14 Proof By binomial theorem we may denote b (i + a + λ) b = s k i k, (26) k=0 d d P(i + b) = p kb i k, Q(i + b) = q kb i k b Z, k b =0 k b =0 where p b, q b, s k s are independent of i.

15 Proof Thus and similarly ad K=0 [P(i + λ)] a = p k0 p ka 1 i k0 i k a 1 k k a 1 =K = ad K=0 P K i K, (27) [Q(q a+i+λ )] l/d a 1 = where P K, Q K s are independent of i. ( l/d a 1)d K=0 Q K i K, (28)

16 Proof So, (a+ l/d a 1)d+b r N = α a+λ j and finally K=0 ( l/d 1)d+b r N = α a+λ j K=0 k+k 1 +K 2 =K k+k 1 +K 2 =K where 0 K ( l/d 1)d + b l 1. s k P K1 Q K2 ml i=0 σ i α i ji k+k 1+K 2 (29) ml s k P K1 Q K2 σ i i K αj, i (30) i=0

17 Proof Hence, by using the property ml i=0 for j {1,..., m}; k {0,..., l 1} we have σ i i k α i j = 0 (31) r N = r ml+λ+a = 0 0 a l/d 1. (32) NOTE: The crucial point in the proof is formula (23) which in fact shows how to find the right denominator polynomials.

18 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 aq n 1 ), n Z + (a) n = (a; q) n = (1, a) n = (1 a)(1 aq) (1 aq n 1 ) (33) q-hypergeometric (basic) series (q) n = (q; q) n = (1 q)...(1 q n ) ( a1,..., a ) A AΦ B t = b 1,..., b B n=0 (a 1 ) n...(a A ) n (q) n (b 1 ) n...(b B ) n t n. (34)

19 q-world The q-binomial coefficients are defined by [ n ] = k (q; q) n (q; q) k (q; q) n k (35) q-binomial theorem [2, p. 490, Corollary (c)]: (b, a; q) n = n k=0 [ n k ] q (k 2) b n k ( a) k n N (36)

20 Padé approximations/q-world Stihl [Arithmetische Eigenschaften spezieller Heinescher Reihen. Math. Ann. 268, (1984)] Let l, m Z + and α = t (α 1,..., α m ) be given and define σ q,i = σ q,i (l, α) by m ml (α t, w; q 1 ) l = σ q,i w i (37) Then t=1 i=0 ml i=0 for all j {1,..., m}; k {0,..., l 1}. σ q,i (α j q k ) i = 0 (38)

21 Padé approximations/q-world Moreover, by q-binomial theorem, σ q,i = ( 1) i q m(l 2) Σq,ml i (39) holds with Σ q,h = Σ q,h (l, α) = i i m=h [ l i 1 ] [ ] l i m q (i 1 2 )+...+( im 2 ) α i 11 α im m. (40)

22 Padé approximations/q-world Define a operator J = J t by Jf (t) = f (qt) and denote again d = max{deg P(y), deg Q(y)}. Stihl constructed the following explicit type II Padé approximations in variable t for the d series J b F (t), 0 b d 1 at m points.

23 Padé approximations/q-world Let b, l, m, λ N, 0 b < d = max{deg P(y), deg Q(y)} and choose m numbers α 1,..., α m. Put σ q,i = σ q,i (l, α) and ml ml B l,λ (t) = t h b l,λ,h = t ml i [Q; q] i+λ+ l/d 1 σ q,i (41) [P; q] i+λ h=0 i=0 Then B l,λ (t)j b F (α j t) A l,λ,b,j (t) = R l,λ,b,j (t), (42) holds with the parameters [ml, ml + λ 1, ml + l/d + λ]. (43)

24 Padé approximations/q-world This means that the polynomials B l,λ (t) are Padé approximant denominators in variable t for the functions F b,j (t) = J b F (tα j ), b = 0, 1,..., d 1; j = 1,..., m. The linear independence results by applying the above explicit Padé approximations of q-hypergeometric series restrict usually to the following class of functions F (t) = with q < 1. n=0 q m(n 2) (q) n (a 1 ) n...(a l ) n t n, m 2, l m 2 (44)

25 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 Padé approximations. q-analogues of exponential series: E q (z) = n=0 1 (q; q) n z n, Ê q (z) = A q-analogue of Euler s divergent series: D q (t) = n=0 q (n 2) (q; q) n z n (45) (q; q) n t n. (46) n=0

26 Arithmetic of q-series Amou M., André Y., Bertrand D., Bézivin, Borwein P., Bundschuh P., Duverney D., Katsurada M., Nesterenko Yu., Nishioka K., Prévost M., Rivoal T., Stihl Th., Shiokawa I., Waldscmidt M., Wallisser R., Väänänen K., Zudilin W.

27 q-world numbers p-adic, p P: n=1 n=1 i=1 l=0 p n 1 p n, (47) p n, (48) n 1 ± p i p l2 l j=1 (1 ± pj ) 2 (49)

28 q-world numbers n=1 1 + p p 2 p 3 (50) (1 + kp n ), k = 1,..., p 1, (51) n=1 p n n (1 + kp i ), k = 1,..., p 1, (52) i=1

29 q-world numbers Real, p Z \ {0, ±1}: n=1 n=1 i=1 l=0 1 1 p n, (53) 1, (54) n 1 ± p i 1 l j=1 (1 ± pj ) 2, (55)

30 q-world numbers n=1 1 + p 1 p 2 p , (56) (1 + kp n ), k = 0, 1,..., p 1, (57) n=1 p n n (1 + kp i ), k = 0, 1,..., p 1. (58) i=1

31 q-world numbers F 1 + F 3 + F , (59) F 2 + F 4 + F , (60) 1, F an+b (61) n=0 n=0 1 L an+b, (62) where a, b, Z +, F n and L n are the Fibonacci and Lucas numbers, respectively; F 0 = 0, F 1 = 1, L 0 = 2, L 1 = 1.

32 q-world/an analogue of Euler s divergent series Euler s divergent series: n!t n. (63) n=0 A q-analogue of Euler s divergent series: D q (t) = (q; q) n t n. (64) n=0 a special case of D b (t) = (b; q) n t n (65) n=0

33 Modified Maier-Stihl/q-world In the following we will present explicit Padé type approximations in variable t for the q-exponential series First, we have F x (t) = k=0 F x (t) = D x (t) = q (k 2) ( tx) k (t; q) k+1. (66) (x; q) n t n (67) n=0

34 Modified Maier-Stihl/q-world Define q-factorial polynomials n 1 n (x; q) n = (1 xq h ) = s(n, k)x k (68) h=0 k=0 and set s(n, k) = 0, when k < 0 or n < k. Then n s(n, k)( x) n k = q (n 2) x n n N (69) k=0 which plays an essential role in the following modification.

35 Modified Maier-Stihl/q-world Define the following polynomials with ml B l,ν (t) = b l,ν,h (t)t h, (70) h=0 b l,ν,h (t) = ( 1) ml h q (ml+ν 2 ) ( ml+ν h 2 ) (t; q)ml+ν h Σ q,h (71)

36 Modified Maier-Stihl/q-world Then by (68) with b l,ν,h = q m(l 2) B l,ν (t) = ml i+f =H 0 f i+ν ml+ν ml+ν H=0 b l,ν,h t H (72) q (ml+ν 2 ) ( i+ν 2 ) s(i + ν, f )σq,i (73)

37 Modified Padé approximations Now we get the following modified Padé 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 B l,ν (t)d αj (t) + A l,ν,j (t) = L l,ν,j (t) (74) with [ml + ν, ml + ν 1, (m + 1)l + ν] (75) By (75) we have a diagonal type Padé approximation with the free parameter ν.

38 Modified Maier-Stihl/q-world Further where and A l,ν,j (t) = a l,ν,j,n = ml+ν 1 N=0 H+n=N a l,ν,j,n t N, (76) b l,ν,h ( α j ; q) n, (77)

39 Modified Maier-Stihl/q-world L l,ν,j (t) = t (m+1)l+ν q (ml+ν 2 )+m( l 2)+νl ( α j ; q) l α ν j S l,ν,j (t). (78) where with S l,ν,j (t) = s l,ν,j,k t k (79) k=0 s l,ν,j,k = q kν ( α j q l ; q) k m t=1 (α t, α j q k+1 ; q) l. (80)

40 Modified Maier-Stihl/Proof The expansion holds with B l,ν (t)d αj (t) = r N t N, (81) N=0 r N = b l,ν,h ( α j ; q) n. (82) H+n=N

41 Modified Maier-Stihl/Proof Set N = ml + ν + a 0 a l 1 then r N = q m(l 2) ml i+ν q (ml+ν 2 ) ( i+ν 2 ) s(i + ν, f )σq,i ( α j ; q) i+ν f +a = i=0 f =0 q m(l 2) ml σ q,i ( α j ; q) a q (ml+ν 2 ) ( i+ν 2 ) i=0 i+ν s(i + ν, f )( α j q a ; q) i+ν f (83) f =0

42 Modified Maier-Stihl/Proof Here the inner f -sum is evaluated by (69) and so ml r ml+ν+a = q m(l 2)+( ml+ν 2 ) ( αj ; q) a σ q,i (α j q a ) i+ν = q m(l 2)+( ml+ν 2 )+aν ( α j ; q) a α ν j for any 0 a l 1. i=0 m (α t, α j q a ; q 1 ) l = 0 (84) t=1

43 Modified Maier-Stihl/Proof Next we consider the case a = l + k, k N. Then r N = r (m+1)l+ν+k = q m(l 2)+( ml+ν 2 )+(l+k)ν ( α j ; q) l+k α ν j m (α t, α j q k+1 ; q) l. (85) t=1

44 Modified Maier-Stihl Note that by Stihl ml C(x, t) = (xt) h (t; q) (m+1)l+λ h (86) h=0 ( 1) ml h q (ml+λ+1 2 ) ( ml+λ h+1 2 ) Σq,h is a denominator polynomial in variable x and ml B l,ν (t) = (tx) h (t; q) ml+ν h (87) h=0 ( 1) ml h q (ml+ν 2 ) ( ml+ν h 2 ) Σq,h is a denominator polynomial in variable t for q (k 2) ( tx) k F x (t) =. (88) (t; q) k+1 k=0

45 Modified Maier-Stihl By using the above modified approximations we may prove linear independence measures for certain values of the functions D a (z) and E a (z), which can be regarded as q-analogues of Euler s divergent series and the usual exponential series. For the q-exponential function E q (z), our result asserts the linear independence (over any number field) of the values 1 and E q (α j ) (j = 1,..., m) together with its measure having the exponent ω = O(m), which sharpens the known exponent ω = O(m 2 ) obtained by a certain refined version of Siegel s lemma Amou, Matala-aho, Väänänen [On Siegel-Shidlovskii s theory for q-difference equations. Acta Arith. 127, (2007)]

46 Modified Maier-Stihl Let p be a prime number. Then we have the linear independence of the p-adic numbers (1 + kp n ), k = 0, 1,..., p 1 (89) n=1 over Q with a measure having the exponent ω < 2p (90)

47 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 K v with a local degree κ v = [K v : Q v ]. 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.

48 Valuations, heights Further, the notation κv /κ α v = α v, κ v = [K v : Q v ], (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 (α 1,..., α m ) K m is given by H(α) = v α v, α v = max i=1,...,m {1, α i v }. (95)

49 Valuations, heights Further, for any place v of K, and q K, q v 1, we define the characteristic λ by λ = λ q = log H(q) log q v. (96)

50 Modified Maier-Stihl/Applications To state our results we denote ω = ω q = u 0 u 0 + λ q s 0, (97) where s 0 = m 2 + m + m m 2 + m, (98) u 0 = m 2 + m + (m + 1) m 2 + m. (99) Now we fix a place v of K throughout the following.

51 Modified Maier-Stihl/Applications Let m Z + be arbitrary, a, q, α 1,..., α m K, and q v < 1. Denote by f (t) each of the functions D t (a), E q (t), (1 tq n ) (100) n=0 and assume a / q N, α i / q N, α i / α j q Z for all i j, (101) ( ) m + < λ q 1. (102) m 2 + m

52 Modified Maier-Stihl/Applications Then the numbers 1, f (α 1 ),..., f (α m ) belonging to K v are linearly independent over K. Further, there exist positive constants c, d, H 0 depending on a and α i such that k 0 + k 1 f (α 1 ) k m f (α m ) v > c H ωκ/κv +d(log H) 1/2 (103) for all k = t (k 0, k 1,..., k m ) K m+1 \ {0} with H = max(h(k), H 0 ).

53 Modified Maier-Stihl/Applications Here we note that λ q 1 always holds for q v < 1, and the following cases in particular assert λ q = 1: 1. K = I, v is the infinite place of K, and 1/q Z K ; 2. K = Q, v = p P, and q = p l, 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 m 2 + m < 2m + 2 (104) and in general we have ω = O(m),too.

54 Modified Maier-Stihl/Applications In the cases 1 3 we have L 0 + L 1 f (α 1 ) L m f (α m ) p > c H 2m+2+d(log H) 1/2 (105) for all L = t (L 0, L 1,..., L m ) Z m+1 \ {0} with H = max ( L i, H 0 ). i=0,...,m

55 Modified Maier-Stihl/Applications Let p P. Then in each of the four sets (1 + kp n ), k = 0, 1,..., p 1, (106) n=1 n=1 p n n (1 + kp i ), k = 0, 1,..., p 1, (107) i=1 (1 + kp n ), k = 0, 1,..., p 1, (108) n=1 p n n=1 i=1 n (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)

56 q-world q-difference equations E q (qz) = (1 z)e q (z), Ê q (z) = (1 + z)ê q (qz). (111) By using (111) one gets the well-known Euler formulae E q (z) = 1 (z), Ê q (z) = ( z). (112)

57 q-identities From (112) we get which implies Ê q (t)e q ( t) = 1 (113) E 1/q,1/q (t)e q,q (qt) = 1 (114) The following connects q-exponentials and q-divergent n=0 q (n 2) α n (qz) n z n = 1 + αzd α,q (qz) (115)

58 Remaider series technique Prévost and Rivoal [Remainder Padé Approximants for the Exponential Function. Constr. Approx. 25, (2007)] presented new Padé type approximations for the classical exponential series using the remainder series technique, invented in Prévost [A new proof of the irrationality of ζ(2) and ζ(3) using Padé approximants. J. Comput. Appl. Math. 67, (1996)]. To be more precise, they constructed approximations in variable t for the exponential remainder series Φ xαj (t) = k=0 (xα j ) k (1 1/t) k, j = 1,..., m (116)

59 Remaider series technique/modified Maier It is interesting that the above series Φ x (t) and D x (t) share a formal similarity. Namely and Φ x (t) = k=0 ( tx) k [1 ty] k+1 = k=0 x k (1 1/t) k ; (117) [1 ty] k+1 = (1 t 0)(1 t 1)(1 t 2) (1 t k) (118) D x (t) = k=0 q (k 2) ( 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 tq 0 )(1 tq 1 )(1 tq 2 ) (1 tq k ) (120)

60 Remaider series technique/modified Maier By applying the above modified Maier s method we may give another proof for the new Padé approximations for the exponential remaider series Φ x (t) = k=0 ( tx) k [1 ty] k+1 = k=0 x k (1 1/t) k. (121) given by Prévost and Rivoal. Prévost and Rivoal showed that Φ x (t) = T n ( x)t n, (122) n=0 where T n (x) are Touchard polynomials

61 Remaider series technique/modified Maier n T n (x) = S 2 (n, k)x k (123) k=0 defined with Stirling numbers of the second kind S 2 (n, k). Further, In the following s 1 (n, k) denote the Stirling numbers of first kind. So our aim is to construct explicit simultaneous Padé type approximations in variable t for the series Φ αj (t) = T n (α j )t n, j = 1,..., m. (124) n=0

62 Remaider series technique/modified Maier We now define ml B l,ν (t) = b l,ν,h (t)t h, (125) h=0 with b l,ν,h (t) = ( 1) ml h [1 ty] ml+ν h Σ h (α), (126) Then with b l,ν,h = B l,ν (t) = ml i+f =H 0 f i+ν ml+ν ml+ν H=0 b l,ν,h t H, (127) s 1 (i + ν, i + ν f )σ i (α) (128)

63 Remaider series technique/modified Maier Let l, ν, m N and choose m numbers α 1,..., α m. Then B l,ν (t)φ αj (t) + A l,ν,j (t) = L l,ν,j (t) (129) with [ml + ν, ml + ν 1, (m + 1)l + ν] (130) give a diagonal type Padé approximation of the second kind with a free parameter ν.

64 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) Thus a k (1/a) k = ml+ν H=0 k s 1 (k, i)a k i = i=0 k s 1 (k, k i)a i i=0 ml B l,ν (t) = t ml i [1 ty] i+ν σ i = t H i=0 ml i+f =H 0 f i+ν ml+ν s 1 (i + ν, i + ν f )σ i = ml+ν H=0 b l,ν,h t H, (132)

65 Remaider series technique/modified Maier We next study the expansion of the product B l,ν (t)φ αj (t) = r N t N, (133) N=0 where Set Then r N = b l,ν,h T n (α j ) (134) H+n=N N = ml + ν + a, 0 a l 1, a N. (135) H = ml i + f, H + n = N. (136)

66 Remaider series technique/modified Maier It follows that n = i + ν f + a and thus ml r N = a+ν+i f e=0 i=0 s 1 (i + ν, i + ν f )T i+ν f +a (α j ) = σ i i+ν f =0 ml i=0 i+ν σ i f =0 s 1 (i + ν, i + ν f )S 2 (a + ν + i f, e)α e j (137) Denote for shortly I = i + ν, K = I f. Then we are led to study the following inner sums

67 Remaider series technique/modified Maier H a = I a+k K=0 e=0 s 1 (I, K)S 2 (a + K, e)α e j = a+i αj e e=0 I K=e a s 1 (I, K)S 2 (a + K, e) = a+i αj e e=0 K=0 I s 1 (I, K)S 2 (a + K, e) (138)

68 Generalized Stirling orthogonality Let a N. For all I, e N we have I s 1 (I, K)S 2 (a + K, e) = C b,d (a)e d δ I,e b (139) K=0 0 b,d a where the numbers C b,d (a) do not depend on I and e.

69 Remaider series technique/modified Maier First H 0 = which by (9) shows I αj e e=0 K=0 I e=0 I s 1 (I, K)S 2 (K, e) = α e j δ Ie = α I j = α i+ν j (140) ml r ml+ν = σ i α i+ν j i=0 = α ν j ml i=0 σ i α i j = 0. (141)

70 Remaider series technique/modified Maier Further I e=0 H 1 = I αj e e=0 K=0 α e j (δ I,e 1 + eδ Ie ) = α I +1 j which by (9) shows ml r ml+ν+1 = σ i α i+ν+1 j (α ν+1 i=0 I s 1 (I, K)S 2 (K + 1, e) = + I α I j = α i+ν+1 j + (i + ν)α i+ν j = + (i + ν)α i+ν j (142) ml ml j + ναj ν ) σ i αj i + αj ν σ i iαj i = 0. (143) i=0 i=0

71 Remaider series technique/modified Maier And in full generality by (139) we have a+i H a = e=0 α e j 0 b,d a C b,d e d δ I,e b = 0 b,d a 0 b,d a 0 b,d a 0 b,d a C b,d a+i αj e e d δ I,e b = e=0 C b,d α I +b j (I + b) d = C b,d α i+ν+b j (i + ν + b) d = C b,d α i+ν+b j d g=o ( ) d i g (b + ν) d g (144) g

72 Remaider series technique/modified Maier which gives d 0 b,d a g=0 ml r ml+ν+a = σ i H a = C b,d α i+ν+b j ( d g i=0 ml )(b + ν) d g σ i αji i g (145) where 0 g a l 1. Hence, again by using (9) we may deduce for any 0 a l 1. i=0 r ml+ν+a = 0 (146)

73 References [1]. Amou M., Matala-aho T., Väänänen K.: On Siegel-Shidlovskii s theory for q-difference equations. Acta Arith. 127, (2007) [2]. Andrews G., Askey R., Roy R.: Special Functions, Encyclopedia of Mathematics and its Applications, 71. Cambridge University Press, Cambridge, [3]. Chudnovsky G. V.: Padé approximations to the generalized hypergeometric functions. I. J. Math. pures et appl. 58, (1979) [4]. Hata M.: Remarks on Mahler s Transcendence Measure for e. J. Number Theory 54, (1995)