Grossone approach to Hutton and Euler transforms
Applied Mathematics and Computation 255 (2015) 36–43
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Grossone approach to Hutton and Euler transforms Vladimir Kanovei a,⇑,1, Vassily Lyubetsky b a b
IITP and MIIT, Moscow, Russia IITP, Moscow, Russia
a r t i c l e
i n f o
a b s t r a c t The aim of this paper is to demonstrate that several non–rigorous methods of mathematical reasoning in the field of divergent series, mostly related to the Euler and Hutton transforms, may be developed in a correct and consistent way by methods of the grossone analysis. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Divergent series Summability Hyperfinite domain
1. Introduction We begin with the following sequence of symbolic transformations with the shift operator, related to summability of divergent series. Shift operator. An arbitrary series
A ¼ a0 þ a1 þ a2 þ a3 þ
ð1Þ
can be rewritten as
A ¼ a0 þ sa0 þ s2 a0 þ s3 a0 þ ¼ ð1 þ s þ s2 þ s3 þ Þ a0 ;
ð2Þ
where s is the shift operator, an operator of yet indefinite mathematical nature, but acting so that sak ¼ akþ1 , — and then as
A ¼
1 a0 ; 1s
ð3Þ
summing up 1 þ s þ s2 þ s3 þ according to the informal equality
1 þ s þ s2 þ s3 þ ¼
1 : 1s
ð4Þ
Hutton transform. Now, let d – 1 and r½d ¼ d þ s. Formally, 1 1s
r½d 1þd
¼
1sþdþs ð1þdÞð1sÞ
¼
1 1þd
þ ð1 þ s þ s2 þ Þ
1 1þd
dsþs2
¼
¼
dþs þ 1þd þ
1 1þd
1þd
þ 11 s þ
ds2 þs3 1þd
r½d 1þd
þ
9 ¼> > = ¼ > > ;
⇑ Corresponding author. 1
E-mail addresses: [email protected] (V. Kanovei), [email protected] (V. Lyubetsky). Partial financial support of Grant RFBR 13-01-00006 acknowledged.
http://dx.doi.org/10.1016/j.amc.2014.06.037 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð5Þ
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
37
and we get the Hutton transform ðH; dÞ (Hardy [1] for d ¼ 1) of the original series,
A¼
a0 da0 þ a1 da1 þ a2 da2 þ a3 þ þ þ þ : 1þd 1þd 1þd 1þd
ð6Þ 1
Iterated Hutton, or Euler–Jakimovski transform. Let fdn gn¼1 be an infinite sequence of real numbers dn – 1. Applying transformations ðH; d1 Þ; ðH; d2 Þ; ðH; d3 Þ; . . . – as in (5) and (6) – consecutively, so that the first term of every intermediate series is separated after each iteration, we obtain the series of separated (frameboxed) terms as the final result: 1 1s
¼ ¼ ¼ ¼
¼
9 > > > > > > > d þ s d þ s > 1 1 1 1 2 > þ þ ¼ > 1þd1 1þd1 1þd2 1þd2 1s > > > > > ðd1 þsÞðd2 þsÞ 1 d1 þs 1 > > þ þ ¼ > ð1þd1 Þð1þd2 Þ 1s ð1þd1 Þð1þd2 Þ 1þd1 > = ðd1 þsÞðd2 þsÞ d1 þs 1 þ ð1þd1 Þð1þd2 Þ þ ð1þd1 Þð1þd2 Þð1þd3 Þ þ > 1þd1 > > > > ðd1 þsÞðd2 þsÞðd3 þsÞ 1 > þ ð1þd ¼ > > 1 Þð1þd2 Þð1þd3 Þ 1s > > > ... ... ... > > > 1 > X > ðd1 þsÞðd2 þsÞ...ðdk þsÞ > > : > ð1þd1 Þð1þd2 Þ...ð1þdk Þð1þdkþ1 Þ ; 1 1þd1
þ
d1 þ s 1 1þd1 1s
¼
ð7Þ
k¼0
Remark 1. The final series is a formal Newton’s interpolation of the function
1 1s
with the nodes d1 ; d2 ; d3 ; . . ..
We conclude that, in the spirit of (3),
A¼
1 X 1 ðd1 þ sÞðd2 þ sÞ . . . ðdk þ sÞ a0 ¼ a0 ; 1s ð1 þ d 1 Þð1 þ d2 Þ . . . ð1 þ dk Þð1 þ dkþ1 Þ k¼0
ð8Þ
where each polynomial Pk ðsÞ ¼ ðd1 þ sÞðd2 þ sÞ . . . ðdk þ sÞ formally acts on a0 in accordance with the basic equalities
sn a0 ¼ an . Remark 2. Transformation (7) and (8) was explicitly introduced by Jakimovski [2] (as ½F; dn ) based on a series of earlier studies. Yet most notably, the whole idea of iterated transformation with separation of first terms of intermediate series belongs to Leonhard Euler, Institutiones Calculi Differentialis, Part II, Section 10 – see a discussion in Hardy [1, Section 2.6]. This is why we call it the Euler–Jakimovski transformation here. The summability method based on the Euler–Jakimovski transformation works, pending appropriate choice of dn , for rapidly divergent oscillating series like 0! 1! þ 2! 3! þ . See [3] for further references. h
2. Regression: some linear transformations The transformations considered above can be represented by the following infinite matrices:
0
1
B Bd 1 B B0 HðdÞ ¼ 1 þ dB B @0 ... 0
1 D0
0
0
0
...
1
C ...C C d 1 0 ...C C; C 0 d 1 ...A ... ... ... ...
1
B B d1 B D1 B 1 d d Eðfdn gn¼1 Þ ¼ B B D1 22 B B d1 d2 d3 @ D3 ...
0
0
0
d
1
0
0
d
1
0
...
1
C ...C C 0 d 1 ...C C; C 0 0 d ...A ... ... ... ... ...
B B0 B SðdÞ ¼ B B0 B @0
0
0
0
0
1 D1
0
0
0
d1 þd2 D2
1 D2
d1 d2 þd1 d3 þd2 d3 D3
d1 þd2 þd3 D3
...
...
...
1
C ...C C C 0 0 ...C C; C C 1 0 ...A D3 ... ... ...
where Dk ¼ ð1 þ d1 Þð1 þ d2 Þ . . . ð1 þ dkþ1 Þ – so that
½Ek ¼
1 Sðd1 Þ Sðd2 Þ Sðdk Þ ... ; 1 þ dkþ1 1 þ d1 1 þ d2 1 þ dk 0
k ¼ 0; 1; 2; . . . ;
where ½Mk ; k ¼ 0; 1; 2; . . ., is the kth row of any matrix M.
ð9Þ
38
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
h¼ That is, let ~ a ¼ ha0 ; a1 ; a2 ; . . .i, ~
D
a0 da0 þa1 ; 1þd 1þd
E D E a0 d1 a0 þa1 1 þa2 be the infinite vectors representing ; da1þd ; . . . , and ~ ; ; . . . e ¼ 1þd ð1þd Þð1þd Þ 1 1 2
the sums in the right-hand parts of equalities resp. (1), (6) and (8). Then formally 1 ~ a and ~ a: e ¼ Eðfdn gn¼1 Þ ~ h ¼ HðdÞ ~
ð10Þ
3. The grossone approach The transformations above and according equalities (2)–(8) obviously do not look rigorous in any way, in particular, since the nature of the ‘‘operator’’ s is not clear. Meanwhile, methods related to hyperreal fields (see [5,6] on modern trends in this direction) were applied in [4] to give a precise meaning to transformations (2)–(6). The goal of this paper is to extend this study to more complicated equalities (7) and (8) of the Euler–Jakimovski transformation, by methods of the grossone analysis. The reader is recommended to follow this paper in connection with [4]. Remark 3. Whilst the foundations of the ‘‘grossone’’ paradigm of Sergeyev [8] remain work in progress [7,9,10], we will stick to a version introduced in [11, Section 1.2]. It includes some key features of the paradigm, especially suited for dealing with infinite sequences and series. It sees the grossone unit r is an infinite integer, divisible by any finite integer. h We replace series (1) by a sum of the form
A ¼ a0 þ a1 þ a2 þ þ ar :
ð11Þ
Sums of this type, that is, with r or another grossone-based infinite quantity, were earlier considered in [11]. Definition 4 (shadows, Definition 1 in [4]). A standard series a0 þ a1 þ a2 þ is called the shadow of a hyperfinite sum a0 þ a1 þ þ ar iff ak ak for all standard k, where
x y iff
jx yj is infinitesimal:
A standard infinite matrix X with terms xkl is called the shadow of a hyperreal ðr þ 1Þ ðr þ 1Þ matrix Y iff xkl ykl for all standard k; l. h In this paper, we will introduce certain linear transformations of infinite sums like (11) by matrices of dimension ðr þ 1Þ ðr þ 1Þ, which correspond to the Euler–Jakimovski transformation, and whose action does not change the sum value. We also show that, provided some conditions are satisfied, shadows of transformed infinite sums are equal to the results of standard transformations of their shadows. It must be said that the Hutton and Jakimovski summation methods belong to a wide variety of linear summation methods (see [1]). Our choice of these methods to be considered in this paper is based not on their special position in this variety (on the contrary, the Hutton summability method is dominated by some other methods, see, e.g. [1], comments to Chapter 1), but rather because of their transparent connection with the shift operator, exploited in Section 1. It is a challenging problem to consider some other summability methods on the base of the same methodology, of course. 4. The shift matrix and the Hutton r-transform In this section, we present some definitions and results of [4], mainly related to the Hutton transform, which are necessary for understanding of the main content of the paper. We are going to consider (11) in the form
A ¼ a0 þ a1 1ð1Þ þ a2 1ð2Þ þ þ ar 1ðrÞ ;
ð12Þ
nðkÞ ¼ ðn x0 Þ ðn x1 Þ . . . ðn xk1 Þ;
ð13Þ
where
and separately; nð0Þ ¼ 1;
which is implied by Newton’s interpolation theorem with nodes x0 ; x1 ; . . . ; xr (Proposition 2 in [4], called the nonstandard Taylor expansion there.) Blanket Assumption 5. It will be assumed that: 1 : an internal sequence of hyperreals xk ; 0 6 k 6 r, is fixed; 2 : nðkÞ ¼ ðn x0 Þ ðn x1 Þ . . . ðn xk1 Þ, in accordance with (13); 3 : hyperreals x0 ; . . . ; xr are pairwise different and xr ¼ 1. h Working in these assumptions and making use of the matrices
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
0
0
B0 B B B0 T ¼ B B... B B @0 0
1
0
0
... 0
0
1
0
...
0
0
1
...
... ... ... ...
1
0
1 B B0 B U¼B B0 B @... 0
0 C C C 0 C C; ...C C C 1 A
0
0
0
...
0
0
0
... 0
39
1 ... 0 C ... 0 C C 0 1 ... 0 C C; C ... ... ... ...A 0 0 ... 1
0 1
0 0
of dimension ðr þ 1Þ ðr þ 1Þ, and ðr þ 1Þ-vectors
~ ak ¼ hak ; akþ1 ; . . . ; ar ; 0; 0; . . . ; 0i; |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
k ¼ 0; 1; 2; . . . ; r;
ð14Þ
k zeros
we demonstrated in [4] that
a0 ¼ ½Tk ~ a0 0 ¼ ½~ ak 0 ¼ ak : ½Tk 0 ~
ð15Þ
(Recall that ½Mn denotes the nth row of a matrix M and rows are enumerated from top to bottom starting with 0 as the index of the top row.) Then equality (12) takes the form
h i a0 ; A ¼ U þ T 1ð1Þ þ T2 1ð2Þ þ þ Tr 1ðrÞ ~
ð16Þ
0
considered as the r-counterpart of (2). We proved in [4] that moreover
h
U þ T 1ð1Þ þ T2 1ð2Þ þ þ Tr 1ðrÞ
i
¼
0
1 ; 1T 0
ð17Þ
where
0
1 B0 B B B0 B B 1 ¼ U þ XT ¼ B 0 B B... B B @0 0
x0
0
1
x1
0
... 0
0
1
C C C C 0 1 x2 . . . 0 0 C C 0 0 1 ... 0 0 C C ... ... ... ... ... ... C C C 0 0 0 ... 1 xr1 A 0 0 0 ... 0 1 0
... 0
0
(the modified unit matrix of dimension ðr þ 1Þ ðr þ 1Þ, and
0
x0
B0 B B B0 X ¼ B B0 B B @... 0
0
0
0
... 0
0
1
C C C C of dimension C 0 0 x3 . . . 0 0 C C ðr þ 1Þ ðr þ 1Þ: C ... ... ... ... ... ... A x1 0
0 x2
0 0
... 0 ... 0
0 0
0
0
0
... 0
xr
so that the equality (17) is the r-counterpart of (3). But (17) has the precise meaning of a sum and a complete proof which does not depend on any special assumption about the nature of the initial series (12). a0 and, using (15) and (16), obtain the following counterpart of (4): Now we multiply both sides of (17) with ~
a0 þ a1 1ð1Þ þ a2 1ð2Þ þ þ ar 1ðrÞ ¼ A ¼
1 ~ a0 : 1T 0
ð18Þ
Thus the analogy between (1)–(4) on the one hand and (12), (16), (18), (17) on the other, is based on the identification of s with T; 1 (in the denominator 1 s) with 1, and on the adjoining of factors 1ðkÞ and taking the 0th row. Following this line, we developed the Hutton transform from this standpoint in [4]. Namely, given a parameter d – 1, we defined the Hutton rmatrix
0
1 þ dx0
B Bd dT þ dX þ U 1 B B0 HðdÞ ¼ ¼ 1þd 1þdB B @... 0 of dimension ðr þ 1Þ ðr þ 1Þ, where
0
... 0
0
1 þ dx1
... 0
0
d
... 0
0
...
... ... ...
0
... d
1 þ dxr
1 C C C C C C A
40
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
0
0
0
0
... 0
0
1
C B 0 0 ... 0 0 C B1 C B B 1 0 ... 0 0 C T ¼ B0 C C B @... ... ... ... ... ...A 0 0 0 ... 1 0
of dimension
ðr þ 1Þ ðr þ 1Þ
is the transpose of T. Then, for any ðr þ 1Þ-vector ~ h ¼ HðdÞ ~ a ¼ ha0 ; a1 ; . . . ; ar i, the product ~ a is still a ðr þ 1Þ-vector, defined by
h0 ¼
1 þ dx0 a0 ; 1þd
and hk ¼
dak1 þ ð1 þ dxk Þ ak 1þd
for 1 6 k 6 r:
a converts the hyperfinite sum (12) to Thus the action of HðdÞ on ~ r X
hk 1ðkÞ
¼
1þdx0 1þd
a0 þ
k¼0
r X
dak1 þð1þdxk Þ ak 1þd
k¼1
¼
r X
9 > > > 1ðkÞ ¼ > > =
ð19Þ
> > > > > ;
a: ½HðdÞk 1ðkÞ ~
k¼0
Theorem 6 (proved in [4]). Assume that the series in (1) is the shadow of the sum in (12), and xk 0 for all standard k. Then the standard matrix HðdÞ of Section 2 is the shadow of HðdÞ and the series in (6) is the shadow of the sum in the 2nd line of (19). P P Moreover, if ~ h 1ðkÞ ¼ r a 1ðkÞ , so that the Hutton r-transform does not change the value of h ¼ HðdÞ ~ a as above, then r k¼0 k k¼0 k sums of the form (12). Proof (sketch). The first claim is rather obvious. To check the moreover claim note that the coefficient of each ak in the right– hand side of the first line of (19) is, indeed, equal to 1ðkÞ , in accordance with (12). h
5. A technical theorem Here we prove a theorem (Theorem 7) which generalizes an earlier result in [4, 5.2]. We will need this result in Section 7. We first define the matrix
0
d 1 þ dx0 B 0 d B B B 0 0 SðdÞ ¼ d 1 þ T ¼ B B... ... B B @ 0 0 0 0
0
1
0
0
...
0
1 þ dx1
0
...
d
1 þ dx2
...
...
...
...
0 0
0 0
... ...
C C C C 0 0 C C ... ... C C d 1 þ dxr1 A 0 d 0
0
of dimension ðr þ 1Þ ðr þ 1Þ. Recall that T is the transpose of T. Put
0
X ¼ Tk X Tk
k
xk 0 B 0 x B kþ1 B 0 ¼ B B 0 B 0 @ 0 ... ...
1
0
0
...
0
0
C ...C C ...C C C ...A ...
xkþ2
0
0
xkþ3
...
...
a diagonal matrix of dimension ðr þ 1Þ ðr þ 1Þ, with xk ; xkþ1 ; . . . ; xr ; ½0 0; ½0 . . . ; 0 (k zeros) on the diagonal. Then we define, for k ¼ 0; 1; . . . ; r, k
1 ¼ U þ k X T;
k
SðdÞ ¼ d k 1 þ T;
and
k
1
FðxÞ ¼ ðk 1 xTÞ :
In particular, 0 X ¼ X, 0 1 ¼ 1; 0 SðdÞ ¼ SðdÞ, and 0 FðxÞ ¼ ð1 xTÞ1 . Moreover, in the assumptions of Blanket Agreement 5, if xn 0 for all standard n, and d; k are limited, then the shadow of k SðdÞ is equal to the matrix SðdÞ in Section 2, the shadow of X is the infinite (in the usual sense) null matrix, and the shadow of k 1 is the infinite unit matrix. Theorem 7. For any k, we have
1 1 þ dxk 1 x0 k 1 1 þ dxk 1 xk 1 ¼ ½U þ SðdÞ ¼ ½U þ 0 0 k1 T k1 T 1þd 1þd 1þd 1 þ d kþ1 1 T 0 0 0
kþ1
SðdÞ:
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V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
Proof. We first observe that ½U0 ¼
1
1þdxk k ½ 1 1þd
k k T0 þ 1x ½ SðdÞ0 . Multiplying both sides by ðk 1 TÞ , we obtain 1þd
1 1 1 þ dx0 1 x0 k 1 ¼ ½U ¼ ½U þ SðdÞ ; 0 k 0 k1 T k1 T 1T 1þd 1þd 0 0
which is the first equality of the theorem. To establish the second equality, it is enough to show that
ðkþ1 1 TÞ k SðdÞ ¼
kþ1
SðdÞ ðk 1 TÞ;
or equivalently
ðkþ1 1 TÞ ðd k 1 þ TÞ ¼ ðd kþ1 1 þ TÞ ðk 1 TÞ; or equivalently, after opening the brackets and simplifying, kþ1
1 T d T k 1 ¼ T k 1 d kþ1 1 T;
that is, ð1 þ dÞ kþ1 1 T ¼ ð1 þ dÞ T k 1, or
kþ1
1 T ¼ T k 1, or
ðU þ kþ1 X TÞ T ¼ T ðU þ k X TÞ; or equivalently (as U T ¼ T UÞ):
0
0 B B0 B kþ1 k XT ¼ T X ¼ B B0 B @0 ... as required.
kþ1
X T ¼ T k X. Yet by definition
1
xkþ1
0
0
0
...
0
xkþ2
0
0
C ...C C ...C C; C ...A ...
0
0
xkþ3
0
0 ...
0 ...
0 ...
xkþ4 ...
h
6. Euler–Jakimovski r-transform In this section we concentrate on the r-interpretation of the Euler–Jakimovski transform. The plan is roughly the same as in the previous section. We define a r-version E of the Euler–Jakimovski matrix E ¼ Eðfdn gÞ of Section 2, prove that the latter is the shadow of the former (under certain conditions) and that its action commutes with the shadow operation (this will require a bit more efforts than in the case of the Hutton transform), prove that E can be obtained by a certain simulation of Eqs. (7) and (8), and then show, in Section 7, that a suitable reiteration of the Hutton r-transform of Section 4 leads to E as well. Blanket Assumption 8. A sequence of hyperreals xk ; k 6 r, satisfying conditions of Blanket Assumption 5, a hyperfinite r
a ¼~ a0 continue to be fixed. Also fix an internal sequence fdn gn¼1 of hyperreals sum (12), and the corresponding vector ~ dn – 1. h Definition 9 (Euler–Jakimovski r-transform). We set ðkÞ
S ¼ k Sðdk Þ
k1
Sðdk1 Þ . . . 1 Sðd1 Þ
for all k 6 r. Define the Euler–Jakimovski matrix E by the equalities
½Ek ¼
ðkÞ 1 þ dkþ1 xk S 0 ð1 þ d1 Þð1 þ d2 Þ . . . ð1 þ dk Þð1 þ dkþ1 Þ
ð20Þ
e of ~ a (~ e;~ a are r þ 1-vectors) is defined by — for all k; 0 6 k 6 r. (Compare with (9).) The Euler–Jakimovski r-transform ~ ~ e ¼ E~ a. h We are able now to define the r-version r X
ek 1ðkÞ ;
9 > > = e ¼ E~ a ¼ E~ a0 ; > where he0 ; . . . ; er i ¼ ~
k¼0 ð1þdkþ1 xk Þ
½ðkÞ S0
so that ek ¼ ð1þd1 Þð1þd2 Þ ... ð1þd
kþ1 Þ
~ a for all k
> > > ;
ð21Þ
P P ðkÞ ðkÞ of (8), similar to (19) in Section 4. The difference between (21) and (19) is that it is not evident that r ¼ r k¼0 ek 1 k¼0 ak 1 — compare with Theorem 6. We will address this question in the next section.To illustrate the definition, we present the first few values of ek :
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V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
e0 ¼ K 0 a0 ; e1 ¼ K 1 ðd1 a0 þ ð1 þ d1 x1 Þ a1 Þ; e2 ¼ K 2 ðd1 d2 a0 þ ðd1 þ d2 þ d1 d2 ðx1 þ x2 ÞÞ a1 þ ð1 þ d1 x2 Þð1 þ d2 x2 Þ a2 Þ; e3 ¼ K 3 ½d1 d2 d3 a0 þ ðd1 d2 þ d2 d3 þ d1 d3 þ d1 d2 d3 ðx1 þ x2 þ x3 ÞÞ a1 þ ðd1 þ d2 þ d3 þ ðd1 d2 þ d2 d3 þ d1 d3 Þðx2 þ x3 Þ þd1 d2 d3 ðx22 þ x2 x3 þ x23 Þ a2 þ ð1 þ d1 x3 Þð1 þ d2 x3 Þð1 þ d3 x3 Þ a3 ; 1þdnþ1 xn 1þd2 x1 1þd3 x2 1 x0 and so on, where K n ¼ ð1þd1 Þð1þd , so that in particular K 0 ¼ 1þd , K 1 ¼ ð1þd , K 2 ¼ ð1þd1 Þð1þd , et cetera. 1þd1 2 Þ ... ð1þdnþ1 Þ 1 Þð1þd2 Þ 2 Þð1þd3 Þ
Proposition 10. Assume that the series in (1) is the shadow of the sum in (12), and xk 0 for limited k. Then the series in (8) is the P shadow of the sum r e 1ðkÞ in (21). k¼0 k Proof. Say that a matrix M is almost triangular iff there is a finite natural number n0 such that l 6 k þ n0 whenever M kl – 0. (M kl is the hk; lith element of M.) Such a matrix can contain nonzero elements above the diagonal only in finitely many consecutive diagonals. The matrices T; T ; U; E; H, and k X; k 1; k Sðdk Þ; ðkÞ S for all standard indices k are almost triangular; the product of a standard number of almost triangular matrices is almost triangular as well. It is also clear that if a standard double-infinite matrix Z is the shadow (see Definition 4) of an almost triangular ðr þ 1Þ ðr þ 1Þ matrix Z, and a standard infinite vector ~ v is the shadow of a ðr þ 1Þ-dimensional vector ~v, then Z ~ v is the shadow of Z~ v. Note that, in the assumptions of Proposition 10, the shadow of every matrix k X (k standard) is the zero matrix. Since the property of being the shadow is preserved under addition and multiplication (for almost triangular matrices), the matrix Sðdk Þ introduced in Section 2 is the shadow of every matrix k Sðdk Þ, and accordingly the product Sðdk Þ Sðdk Þ . . . Sðd1 Þ is the shadow of ðkÞ S (provided k is standard). Therefore E ¼ Eðfdn gÞ is the shadow of E. This completes the proof of Proposition 10. h
7. Euler–Jakimovski matrix via reiteration of Hutton transform We are going to demonstrate here that the hyperfinite sum transform of Section 4, similarly to Eqs. (7) and (8).
Pr
ðkÞ k¼0 ek 1
in (21) can be obtained by reiteration of the Hutton
Theorem 11 (under Blanket Assumption 8). We have
r X ðkÞ ðkÞ 1 1 þ dkþ1 xk ¼ S 01 : ð1 þ d Þð1 þ d Þ . . . ð1 þ d Þ 1T 0 1 2 kþ1 k¼0
1 Proof. Let us apply the equality of Theorem 7 consecutively, starting with 1T , for the indices k ¼ 0; 1; . . . ; r 1 and accordingly values d ¼ d1 ; d2 ; . . . ; dr . The first two steps will be as follows:
1 1 þ d1 x0 1 x0 1 1 ¼ ½U0 þ Sðd1 Þ 1T 0 1 þ d1 1 þ d1 1 1 T 0
1 þ d 1 x0 1 x0 1 þ d2 x1 1 x1 1 1 2 ½U0 þ ½U0 þ Sðd Þ Sðd1 Þ ¼ 2 1 þ d1 1 þ d1 1 þ d2 1 þ d2 2 1 T 0 1 þ d 1 x0 ð1 x0 Þð1 þ d2 x1 Þ ð1 x0 Þð1 x1 Þ 1 ð2Þ ¼ ½U0 þ S: ½U0 1 Sðd1 Þ þ 2 ð1 þ d1 Þð1 þ d2 Þ ð1 þ d1 Þð1 þ d2 Þ 1 T 0 1 þ d1
After r steps, the rightmost term turns out to be
Nrþ1
1 ðrþ1Þ S 1ðrþ1Þ ¼ Nrþ1 ½ U 0 ðrþ1Þ S 1ðrþ1Þ ¼ 0; rþ1 1 T 0
where
Nk ¼
1 ð1 þ d1 Þð1 þ d2 Þ . . . ð1 þ dk Þ
and 1ðkÞ ¼ ð1 x0 Þð1 x1 Þ . . . ð1 xk1 Þ — so that 1ðrþ1Þ ¼ 0 because xr ¼ 1. Therefore, we obtain the equality of the theorem after exactly the number r of ‘‘Hutton’’ steps. h
43
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
Corollary 12. In the notation of (12) of Section 4 and (21) of Section 6, r X
ek 1ðkÞ ¼
k¼0
r X ð1 þ dkþ1 xk Þ ðkÞ S 0 a¼ ak 1ðkÞ ¼ A: ~ ð1 þ d Þð1 þ d Þ . . . ð1 þ d Þ 1 2 kþ1 k¼0 k¼0
r X
Proof. The left and right equalities hold by definition. To prove the middle one, recall that a. h Section 4. It remains to multiply the equality of the theorem by ~
1 1T 0
~ a¼
Pr
ðkÞ k¼0 ak 1
by (18) in
It follows that the Euler–Jakimovski r-transform does not change the value of sums of the form (12). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
G.H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949. A. Jakimovski, A generalization of Lototsky method of summability, Mich. Math. J. 6 (3) (1959) 277–290. V. Kanovei, On Euler’s summation of the series of alternating factorials. Istoriko-matematicheskie issledovaniya, vol. 34, 1993, pp. 8–45 (Russian). V. Kanovei, M. Reeken, Summation of divergent series from the nonstandard point of view, Real Anal. Exch. 21 (2) (1996) 453–477. V. Kanovei, M. Reeken, Nonstandard Analysis: Axiomatically. Springer Monographs in Mathematics, 2004, XVI+408 p. V. Kanovei, V. Lyubetsky, Problems of set theoretic nonstandard analysis, Russ. Math. Surv. 62 (1) (2007) 45–111. G. Lolli, Infinitesimals and infinities in the history of mathematics, a brief survey, Appl. Math. Comput. 218 (2012) 7979–7988. Ya. D. Sergeyev, Arithmetic of infinity, Edizioni Orizzonti Meridionali, 2003, VIII+104 pp. Ya. D. Sergeyev, Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains, Non-linear Anal. Ser. A Theory Methods Appl. 71 (12) (2009) e1688–e1707. [10] Ya. D. Sergeyev, On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function, p-Adic Numbers, Ultrametric Anal. Appl. 3 (2) (2011) 129–148. [11] A. Zhigljavsky, Computing sums of conditionally convergent and divergent series using the concept of grossone, Appl. Math. Comput. 218 (2012) 8064– 8076.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Grossone approach to Hutton and Euler transforms Vladimir Kanovei a,⇑,1, Vassily Lyubetsky b a b
IITP and MIIT, Moscow, Russia IITP, Moscow, Russia
a r t i c l e
i n f o
a b s t r a c t The aim of this paper is to demonstrate that several non–rigorous methods of mathematical reasoning in the field of divergent series, mostly related to the Euler and Hutton transforms, may be developed in a correct and consistent way by methods of the grossone analysis. Ó 2014 Elsevier Inc. All rights reserved.
Keywords: Divergent series Summability Hyperfinite domain
1. Introduction We begin with the following sequence of symbolic transformations with the shift operator, related to summability of divergent series. Shift operator. An arbitrary series
A ¼ a0 þ a1 þ a2 þ a3 þ
ð1Þ
can be rewritten as
A ¼ a0 þ sa0 þ s2 a0 þ s3 a0 þ ¼ ð1 þ s þ s2 þ s3 þ Þ a0 ;
ð2Þ
where s is the shift operator, an operator of yet indefinite mathematical nature, but acting so that sak ¼ akþ1 , — and then as
A ¼
1 a0 ; 1s
ð3Þ
summing up 1 þ s þ s2 þ s3 þ according to the informal equality
1 þ s þ s2 þ s3 þ ¼
1 : 1s
ð4Þ
Hutton transform. Now, let d – 1 and r½d ¼ d þ s. Formally, 1 1s
r½d 1þd
¼
1sþdþs ð1þdÞð1sÞ
¼
1 1þd
þ ð1 þ s þ s2 þ Þ
1 1þd
dsþs2
¼
¼
dþs þ 1þd þ
1 1þd
1þd
þ 11 s þ
ds2 þs3 1þd
r½d 1þd
þ
9 ¼> > = ¼ > > ;
⇑ Corresponding author. 1
E-mail addresses: [email protected] (V. Kanovei), [email protected] (V. Lyubetsky). Partial financial support of Grant RFBR 13-01-00006 acknowledged.
http://dx.doi.org/10.1016/j.amc.2014.06.037 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
ð5Þ
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
37
and we get the Hutton transform ðH; dÞ (Hardy [1] for d ¼ 1) of the original series,
A¼
a0 da0 þ a1 da1 þ a2 da2 þ a3 þ þ þ þ : 1þd 1þd 1þd 1þd
ð6Þ 1
Iterated Hutton, or Euler–Jakimovski transform. Let fdn gn¼1 be an infinite sequence of real numbers dn – 1. Applying transformations ðH; d1 Þ; ðH; d2 Þ; ðH; d3 Þ; . . . – as in (5) and (6) – consecutively, so that the first term of every intermediate series is separated after each iteration, we obtain the series of separated (frameboxed) terms as the final result: 1 1s
¼ ¼ ¼ ¼
¼
9 > > > > > > > d þ s d þ s > 1 1 1 1 2 > þ þ ¼ > 1þd1 1þd1 1þd2 1þd2 1s > > > > > ðd1 þsÞðd2 þsÞ 1 d1 þs 1 > > þ þ ¼ > ð1þd1 Þð1þd2 Þ 1s ð1þd1 Þð1þd2 Þ 1þd1 > = ðd1 þsÞðd2 þsÞ d1 þs 1 þ ð1þd1 Þð1þd2 Þ þ ð1þd1 Þð1þd2 Þð1þd3 Þ þ > 1þd1 > > > > ðd1 þsÞðd2 þsÞðd3 þsÞ 1 > þ ð1þd ¼ > > 1 Þð1þd2 Þð1þd3 Þ 1s > > > ... ... ... > > > 1 > X > ðd1 þsÞðd2 þsÞ...ðdk þsÞ > > : > ð1þd1 Þð1þd2 Þ...ð1þdk Þð1þdkþ1 Þ ; 1 1þd1
þ
d1 þ s 1 1þd1 1s
¼
ð7Þ
k¼0
Remark 1. The final series is a formal Newton’s interpolation of the function
1 1s
with the nodes d1 ; d2 ; d3 ; . . ..
We conclude that, in the spirit of (3),
A¼
1 X 1 ðd1 þ sÞðd2 þ sÞ . . . ðdk þ sÞ a0 ¼ a0 ; 1s ð1 þ d 1 Þð1 þ d2 Þ . . . ð1 þ dk Þð1 þ dkþ1 Þ k¼0
ð8Þ
where each polynomial Pk ðsÞ ¼ ðd1 þ sÞðd2 þ sÞ . . . ðdk þ sÞ formally acts on a0 in accordance with the basic equalities
sn a0 ¼ an . Remark 2. Transformation (7) and (8) was explicitly introduced by Jakimovski [2] (as ½F; dn ) based on a series of earlier studies. Yet most notably, the whole idea of iterated transformation with separation of first terms of intermediate series belongs to Leonhard Euler, Institutiones Calculi Differentialis, Part II, Section 10 – see a discussion in Hardy [1, Section 2.6]. This is why we call it the Euler–Jakimovski transformation here. The summability method based on the Euler–Jakimovski transformation works, pending appropriate choice of dn , for rapidly divergent oscillating series like 0! 1! þ 2! 3! þ . See [3] for further references. h
2. Regression: some linear transformations The transformations considered above can be represented by the following infinite matrices:
0
1
B Bd 1 B B0 HðdÞ ¼ 1 þ dB B @0 ... 0
1 D0
0
0
0
...
1
C ...C C d 1 0 ...C C; C 0 d 1 ...A ... ... ... ...
1
B B d1 B D1 B 1 d d Eðfdn gn¼1 Þ ¼ B B D1 22 B B d1 d2 d3 @ D3 ...
0
0
0
d
1
0
0
d
1
0
...
1
C ...C C 0 d 1 ...C C; C 0 0 d ...A ... ... ... ... ...
B B0 B SðdÞ ¼ B B0 B @0
0
0
0
0
1 D1
0
0
0
d1 þd2 D2
1 D2
d1 d2 þd1 d3 þd2 d3 D3
d1 þd2 þd3 D3
...
...
...
1
C ...C C C 0 0 ...C C; C C 1 0 ...A D3 ... ... ...
where Dk ¼ ð1 þ d1 Þð1 þ d2 Þ . . . ð1 þ dkþ1 Þ – so that
½Ek ¼
1 Sðd1 Þ Sðd2 Þ Sðdk Þ ... ; 1 þ dkþ1 1 þ d1 1 þ d2 1 þ dk 0
k ¼ 0; 1; 2; . . . ;
where ½Mk ; k ¼ 0; 1; 2; . . ., is the kth row of any matrix M.
ð9Þ
38
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
h¼ That is, let ~ a ¼ ha0 ; a1 ; a2 ; . . .i, ~
D
a0 da0 þa1 ; 1þd 1þd
E D E a0 d1 a0 þa1 1 þa2 be the infinite vectors representing ; da1þd ; . . . , and ~ ; ; . . . e ¼ 1þd ð1þd Þð1þd Þ 1 1 2
the sums in the right-hand parts of equalities resp. (1), (6) and (8). Then formally 1 ~ a and ~ a: e ¼ Eðfdn gn¼1 Þ ~ h ¼ HðdÞ ~
ð10Þ
3. The grossone approach The transformations above and according equalities (2)–(8) obviously do not look rigorous in any way, in particular, since the nature of the ‘‘operator’’ s is not clear. Meanwhile, methods related to hyperreal fields (see [5,6] on modern trends in this direction) were applied in [4] to give a precise meaning to transformations (2)–(6). The goal of this paper is to extend this study to more complicated equalities (7) and (8) of the Euler–Jakimovski transformation, by methods of the grossone analysis. The reader is recommended to follow this paper in connection with [4]. Remark 3. Whilst the foundations of the ‘‘grossone’’ paradigm of Sergeyev [8] remain work in progress [7,9,10], we will stick to a version introduced in [11, Section 1.2]. It includes some key features of the paradigm, especially suited for dealing with infinite sequences and series. It sees the grossone unit r is an infinite integer, divisible by any finite integer. h We replace series (1) by a sum of the form
A ¼ a0 þ a1 þ a2 þ þ ar :
ð11Þ
Sums of this type, that is, with r or another grossone-based infinite quantity, were earlier considered in [11]. Definition 4 (shadows, Definition 1 in [4]). A standard series a0 þ a1 þ a2 þ is called the shadow of a hyperfinite sum a0 þ a1 þ þ ar iff ak ak for all standard k, where
x y iff
jx yj is infinitesimal:
A standard infinite matrix X with terms xkl is called the shadow of a hyperreal ðr þ 1Þ ðr þ 1Þ matrix Y iff xkl ykl for all standard k; l. h In this paper, we will introduce certain linear transformations of infinite sums like (11) by matrices of dimension ðr þ 1Þ ðr þ 1Þ, which correspond to the Euler–Jakimovski transformation, and whose action does not change the sum value. We also show that, provided some conditions are satisfied, shadows of transformed infinite sums are equal to the results of standard transformations of their shadows. It must be said that the Hutton and Jakimovski summation methods belong to a wide variety of linear summation methods (see [1]). Our choice of these methods to be considered in this paper is based not on their special position in this variety (on the contrary, the Hutton summability method is dominated by some other methods, see, e.g. [1], comments to Chapter 1), but rather because of their transparent connection with the shift operator, exploited in Section 1. It is a challenging problem to consider some other summability methods on the base of the same methodology, of course. 4. The shift matrix and the Hutton r-transform In this section, we present some definitions and results of [4], mainly related to the Hutton transform, which are necessary for understanding of the main content of the paper. We are going to consider (11) in the form
A ¼ a0 þ a1 1ð1Þ þ a2 1ð2Þ þ þ ar 1ðrÞ ;
ð12Þ
nðkÞ ¼ ðn x0 Þ ðn x1 Þ . . . ðn xk1 Þ;
ð13Þ
where
and separately; nð0Þ ¼ 1;
which is implied by Newton’s interpolation theorem with nodes x0 ; x1 ; . . . ; xr (Proposition 2 in [4], called the nonstandard Taylor expansion there.) Blanket Assumption 5. It will be assumed that: 1 : an internal sequence of hyperreals xk ; 0 6 k 6 r, is fixed; 2 : nðkÞ ¼ ðn x0 Þ ðn x1 Þ . . . ðn xk1 Þ, in accordance with (13); 3 : hyperreals x0 ; . . . ; xr are pairwise different and xr ¼ 1. h Working in these assumptions and making use of the matrices
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
0
0
B0 B B B0 T ¼ B B... B B @0 0
1
0
0
... 0
0
1
0
...
0
0
1
...
... ... ... ...
1
0
1 B B0 B U¼B B0 B @... 0
0 C C C 0 C C; ...C C C 1 A
0
0
0
...
0
0
0
... 0
39
1 ... 0 C ... 0 C C 0 1 ... 0 C C; C ... ... ... ...A 0 0 ... 1
0 1
0 0
of dimension ðr þ 1Þ ðr þ 1Þ, and ðr þ 1Þ-vectors
~ ak ¼ hak ; akþ1 ; . . . ; ar ; 0; 0; . . . ; 0i; |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}
k ¼ 0; 1; 2; . . . ; r;
ð14Þ
k zeros
we demonstrated in [4] that
a0 ¼ ½Tk ~ a0 0 ¼ ½~ ak 0 ¼ ak : ½Tk 0 ~
ð15Þ
(Recall that ½Mn denotes the nth row of a matrix M and rows are enumerated from top to bottom starting with 0 as the index of the top row.) Then equality (12) takes the form
h i a0 ; A ¼ U þ T 1ð1Þ þ T2 1ð2Þ þ þ Tr 1ðrÞ ~
ð16Þ
0
considered as the r-counterpart of (2). We proved in [4] that moreover
h
U þ T 1ð1Þ þ T2 1ð2Þ þ þ Tr 1ðrÞ
i
¼
0
1 ; 1T 0
ð17Þ
where
0
1 B0 B B B0 B B 1 ¼ U þ XT ¼ B 0 B B... B B @0 0
x0
0
1
x1
0
... 0
0
1
C C C C 0 1 x2 . . . 0 0 C C 0 0 1 ... 0 0 C C ... ... ... ... ... ... C C C 0 0 0 ... 1 xr1 A 0 0 0 ... 0 1 0
... 0
0
(the modified unit matrix of dimension ðr þ 1Þ ðr þ 1Þ, and
0
x0
B0 B B B0 X ¼ B B0 B B @... 0
0
0
0
... 0
0
1
C C C C of dimension C 0 0 x3 . . . 0 0 C C ðr þ 1Þ ðr þ 1Þ: C ... ... ... ... ... ... A x1 0
0 x2
0 0
... 0 ... 0
0 0
0
0
0
... 0
xr
so that the equality (17) is the r-counterpart of (3). But (17) has the precise meaning of a sum and a complete proof which does not depend on any special assumption about the nature of the initial series (12). a0 and, using (15) and (16), obtain the following counterpart of (4): Now we multiply both sides of (17) with ~
a0 þ a1 1ð1Þ þ a2 1ð2Þ þ þ ar 1ðrÞ ¼ A ¼
1 ~ a0 : 1T 0
ð18Þ
Thus the analogy between (1)–(4) on the one hand and (12), (16), (18), (17) on the other, is based on the identification of s with T; 1 (in the denominator 1 s) with 1, and on the adjoining of factors 1ðkÞ and taking the 0th row. Following this line, we developed the Hutton transform from this standpoint in [4]. Namely, given a parameter d – 1, we defined the Hutton rmatrix
0
1 þ dx0
B Bd dT þ dX þ U 1 B B0 HðdÞ ¼ ¼ 1þd 1þdB B @... 0 of dimension ðr þ 1Þ ðr þ 1Þ, where
0
... 0
0
1 þ dx1
... 0
0
d
... 0
0
...
... ... ...
0
... d
1 þ dxr
1 C C C C C C A
40
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
0
0
0
0
... 0
0
1
C B 0 0 ... 0 0 C B1 C B B 1 0 ... 0 0 C T ¼ B0 C C B @... ... ... ... ... ...A 0 0 0 ... 1 0
of dimension
ðr þ 1Þ ðr þ 1Þ
is the transpose of T. Then, for any ðr þ 1Þ-vector ~ h ¼ HðdÞ ~ a ¼ ha0 ; a1 ; . . . ; ar i, the product ~ a is still a ðr þ 1Þ-vector, defined by
h0 ¼
1 þ dx0 a0 ; 1þd
and hk ¼
dak1 þ ð1 þ dxk Þ ak 1þd
for 1 6 k 6 r:
a converts the hyperfinite sum (12) to Thus the action of HðdÞ on ~ r X
hk 1ðkÞ
¼
1þdx0 1þd
a0 þ
k¼0
r X
dak1 þð1þdxk Þ ak 1þd
k¼1
¼
r X
9 > > > 1ðkÞ ¼ > > =
ð19Þ
> > > > > ;
a: ½HðdÞk 1ðkÞ ~
k¼0
Theorem 6 (proved in [4]). Assume that the series in (1) is the shadow of the sum in (12), and xk 0 for all standard k. Then the standard matrix HðdÞ of Section 2 is the shadow of HðdÞ and the series in (6) is the shadow of the sum in the 2nd line of (19). P P Moreover, if ~ h 1ðkÞ ¼ r a 1ðkÞ , so that the Hutton r-transform does not change the value of h ¼ HðdÞ ~ a as above, then r k¼0 k k¼0 k sums of the form (12). Proof (sketch). The first claim is rather obvious. To check the moreover claim note that the coefficient of each ak in the right– hand side of the first line of (19) is, indeed, equal to 1ðkÞ , in accordance with (12). h
5. A technical theorem Here we prove a theorem (Theorem 7) which generalizes an earlier result in [4, 5.2]. We will need this result in Section 7. We first define the matrix
0
d 1 þ dx0 B 0 d B B B 0 0 SðdÞ ¼ d 1 þ T ¼ B B... ... B B @ 0 0 0 0
0
1
0
0
...
0
1 þ dx1
0
...
d
1 þ dx2
...
...
...
...
0 0
0 0
... ...
C C C C 0 0 C C ... ... C C d 1 þ dxr1 A 0 d 0
0
of dimension ðr þ 1Þ ðr þ 1Þ. Recall that T is the transpose of T. Put
0
X ¼ Tk X Tk
k
xk 0 B 0 x B kþ1 B 0 ¼ B B 0 B 0 @ 0 ... ...
1
0
0
...
0
0
C ...C C ...C C C ...A ...
xkþ2
0
0
xkþ3
...
...
a diagonal matrix of dimension ðr þ 1Þ ðr þ 1Þ, with xk ; xkþ1 ; . . . ; xr ; ½0 0; ½0 . . . ; 0 (k zeros) on the diagonal. Then we define, for k ¼ 0; 1; . . . ; r, k
1 ¼ U þ k X T;
k
SðdÞ ¼ d k 1 þ T;
and
k
1
FðxÞ ¼ ðk 1 xTÞ :
In particular, 0 X ¼ X, 0 1 ¼ 1; 0 SðdÞ ¼ SðdÞ, and 0 FðxÞ ¼ ð1 xTÞ1 . Moreover, in the assumptions of Blanket Agreement 5, if xn 0 for all standard n, and d; k are limited, then the shadow of k SðdÞ is equal to the matrix SðdÞ in Section 2, the shadow of X is the infinite (in the usual sense) null matrix, and the shadow of k 1 is the infinite unit matrix. Theorem 7. For any k, we have
1 1 þ dxk 1 x0 k 1 1 þ dxk 1 xk 1 ¼ ½U þ SðdÞ ¼ ½U þ 0 0 k1 T k1 T 1þd 1þd 1þd 1 þ d kþ1 1 T 0 0 0
kþ1
SðdÞ:
41
V. Kanovei, V. Lyubetsky / Applied Mathematics and Computation 255 (2015) 36–43
Proof. We first observe that ½U0 ¼
1
1þdxk k ½ 1 1þd
k k T0 þ 1x ½ SðdÞ0 . Multiplying both sides by ðk 1 TÞ , we obtain 1þd
1 1 1 þ dx0 1 x0 k 1 ¼ ½U ¼ ½U þ SðdÞ ; 0 k 0 k1 T k1 T 1T 1þd 1þd 0 0
which is the first equality of the theorem. To establish the second equality, it is enough to show that
ðkþ1 1 TÞ k SðdÞ ¼
kþ1
SðdÞ ðk 1 TÞ;
or equivalently
ðkþ1 1 TÞ ðd k 1 þ TÞ ¼ ðd kþ1 1 þ TÞ ðk 1 TÞ; or equivalently, after opening the brackets and simplifying, kþ1
1 T d T k 1 ¼ T k 1 d kþ1 1 T;
that is, ð1 þ dÞ kþ1 1 T ¼ ð1 þ dÞ T k 1, or
kþ1
1 T ¼ T k 1, or
ðU þ kþ1 X TÞ T ¼ T ðU þ k X TÞ; or equivalently (as U T ¼ T UÞ):
0
0 B B0 B kþ1 k XT ¼ T X ¼ B B0 B @0 ... as required.
kþ1
X T ¼ T k X. Yet by definition
1
xkþ1
0
0
0
...
0
xkþ2
0
0
C ...C C ...C C; C ...A ...
0
0
xkþ3
0
0 ...
0 ...
0 ...
xkþ4 ...
h
6. Euler–Jakimovski r-transform In this section we concentrate on the r-interpretation of the Euler–Jakimovski transform. The plan is roughly the same as in the previous section. We define a r-version E of the Euler–Jakimovski matrix E ¼ Eðfdn gÞ of Section 2, prove that the latter is the shadow of the former (under certain conditions) and that its action commutes with the shadow operation (this will require a bit more efforts than in the case of the Hutton transform), prove that E can be obtained by a certain simulation of Eqs. (7) and (8), and then show, in Section 7, that a suitable reiteration of the Hutton r-transform of Section 4 leads to E as well. Blanket Assumption 8. A sequence of hyperreals xk ; k 6 r, satisfying conditions of Blanket Assumption 5, a hyperfinite r
a ¼~ a0 continue to be fixed. Also fix an internal sequence fdn gn¼1 of hyperreals sum (12), and the corresponding vector ~ dn – 1. h Definition 9 (Euler–Jakimovski r-transform). We set ðkÞ
S ¼ k Sðdk Þ
k1
Sðdk1 Þ . . . 1 Sðd1 Þ
for all k 6 r. Define the Euler–Jakimovski matrix E by the equalities
½Ek ¼
ðkÞ 1 þ dkþ1 xk S 0 ð1 þ d1 Þð1 þ d2 Þ . . . ð1 þ dk Þð1 þ dkþ1 Þ
ð20Þ
e of ~ a (~ e;~ a are r þ 1-vectors) is defined by — for all k; 0 6 k 6 r. (Compare with (9).) The Euler–Jakimovski r-transform ~ ~ e ¼ E~ a. h We are able now to define the r-version r X
ek 1ðkÞ ;
9 > > = e ¼ E~ a ¼ E~ a0 ; > where he0 ; . . . ; er i ¼ ~
k¼0 ð1þdkþ1 xk Þ
½ðkÞ S0
so that ek ¼ ð1þd1 Þð1þd2 Þ ... ð1þd
kþ1 Þ
~ a for all k
> > > ;
ð21Þ
P P ðkÞ ðkÞ of (8), similar to (19) in Section 4. The difference between (21) and (19) is that it is not evident that r ¼ r k¼0 ek 1 k¼0 ak 1 — compare with Theorem 6. We will address this question in the next section.To illustrate the definition, we present the first few values of ek :
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e0 ¼ K 0 a0 ; e1 ¼ K 1 ðd1 a0 þ ð1 þ d1 x1 Þ a1 Þ; e2 ¼ K 2 ðd1 d2 a0 þ ðd1 þ d2 þ d1 d2 ðx1 þ x2 ÞÞ a1 þ ð1 þ d1 x2 Þð1 þ d2 x2 Þ a2 Þ; e3 ¼ K 3 ½d1 d2 d3 a0 þ ðd1 d2 þ d2 d3 þ d1 d3 þ d1 d2 d3 ðx1 þ x2 þ x3 ÞÞ a1 þ ðd1 þ d2 þ d3 þ ðd1 d2 þ d2 d3 þ d1 d3 Þðx2 þ x3 Þ þd1 d2 d3 ðx22 þ x2 x3 þ x23 Þ a2 þ ð1 þ d1 x3 Þð1 þ d2 x3 Þð1 þ d3 x3 Þ a3 ; 1þdnþ1 xn 1þd2 x1 1þd3 x2 1 x0 and so on, where K n ¼ ð1þd1 Þð1þd , so that in particular K 0 ¼ 1þd , K 1 ¼ ð1þd , K 2 ¼ ð1þd1 Þð1þd , et cetera. 1þd1 2 Þ ... ð1þdnþ1 Þ 1 Þð1þd2 Þ 2 Þð1þd3 Þ
Proposition 10. Assume that the series in (1) is the shadow of the sum in (12), and xk 0 for limited k. Then the series in (8) is the P shadow of the sum r e 1ðkÞ in (21). k¼0 k Proof. Say that a matrix M is almost triangular iff there is a finite natural number n0 such that l 6 k þ n0 whenever M kl – 0. (M kl is the hk; lith element of M.) Such a matrix can contain nonzero elements above the diagonal only in finitely many consecutive diagonals. The matrices T; T ; U; E; H, and k X; k 1; k Sðdk Þ; ðkÞ S for all standard indices k are almost triangular; the product of a standard number of almost triangular matrices is almost triangular as well. It is also clear that if a standard double-infinite matrix Z is the shadow (see Definition 4) of an almost triangular ðr þ 1Þ ðr þ 1Þ matrix Z, and a standard infinite vector ~ v is the shadow of a ðr þ 1Þ-dimensional vector ~v, then Z ~ v is the shadow of Z~ v. Note that, in the assumptions of Proposition 10, the shadow of every matrix k X (k standard) is the zero matrix. Since the property of being the shadow is preserved under addition and multiplication (for almost triangular matrices), the matrix Sðdk Þ introduced in Section 2 is the shadow of every matrix k Sðdk Þ, and accordingly the product Sðdk Þ Sðdk Þ . . . Sðd1 Þ is the shadow of ðkÞ S (provided k is standard). Therefore E ¼ Eðfdn gÞ is the shadow of E. This completes the proof of Proposition 10. h
7. Euler–Jakimovski matrix via reiteration of Hutton transform We are going to demonstrate here that the hyperfinite sum transform of Section 4, similarly to Eqs. (7) and (8).
Pr
ðkÞ k¼0 ek 1
in (21) can be obtained by reiteration of the Hutton
Theorem 11 (under Blanket Assumption 8). We have
r X ðkÞ ðkÞ 1 1 þ dkþ1 xk ¼ S 01 : ð1 þ d Þð1 þ d Þ . . . ð1 þ d Þ 1T 0 1 2 kþ1 k¼0
1 Proof. Let us apply the equality of Theorem 7 consecutively, starting with 1T , for the indices k ¼ 0; 1; . . . ; r 1 and accordingly values d ¼ d1 ; d2 ; . . . ; dr . The first two steps will be as follows:
1 1 þ d1 x0 1 x0 1 1 ¼ ½U0 þ Sðd1 Þ 1T 0 1 þ d1 1 þ d1 1 1 T 0
1 þ d 1 x0 1 x0 1 þ d2 x1 1 x1 1 1 2 ½U0 þ ½U0 þ Sðd Þ Sðd1 Þ ¼ 2 1 þ d1 1 þ d1 1 þ d2 1 þ d2 2 1 T 0 1 þ d 1 x0 ð1 x0 Þð1 þ d2 x1 Þ ð1 x0 Þð1 x1 Þ 1 ð2Þ ¼ ½U0 þ S: ½U0 1 Sðd1 Þ þ 2 ð1 þ d1 Þð1 þ d2 Þ ð1 þ d1 Þð1 þ d2 Þ 1 T 0 1 þ d1
After r steps, the rightmost term turns out to be
Nrþ1
1 ðrþ1Þ S 1ðrþ1Þ ¼ Nrþ1 ½ U 0 ðrþ1Þ S 1ðrþ1Þ ¼ 0; rþ1 1 T 0
where
Nk ¼
1 ð1 þ d1 Þð1 þ d2 Þ . . . ð1 þ dk Þ
and 1ðkÞ ¼ ð1 x0 Þð1 x1 Þ . . . ð1 xk1 Þ — so that 1ðrþ1Þ ¼ 0 because xr ¼ 1. Therefore, we obtain the equality of the theorem after exactly the number r of ‘‘Hutton’’ steps. h
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Corollary 12. In the notation of (12) of Section 4 and (21) of Section 6, r X
ek 1ðkÞ ¼
k¼0
r X ð1 þ dkþ1 xk Þ ðkÞ S 0 a¼ ak 1ðkÞ ¼ A: ~ ð1 þ d Þð1 þ d Þ . . . ð1 þ d Þ 1 2 kþ1 k¼0 k¼0
r X
Proof. The left and right equalities hold by definition. To prove the middle one, recall that a. h Section 4. It remains to multiply the equality of the theorem by ~
1 1T 0
~ a¼
Pr
ðkÞ k¼0 ak 1
by (18) in
It follows that the Euler–Jakimovski r-transform does not change the value of sums of the form (12). References [1] [2] [3] [4] [5] [6] [7] [8] [9]
G.H. Hardy, Divergent Series, Clarendon Press, Oxford, 1949. A. Jakimovski, A generalization of Lototsky method of summability, Mich. Math. J. 6 (3) (1959) 277–290. V. Kanovei, On Euler’s summation of the series of alternating factorials. Istoriko-matematicheskie issledovaniya, vol. 34, 1993, pp. 8–45 (Russian). V. Kanovei, M. Reeken, Summation of divergent series from the nonstandard point of view, Real Anal. Exch. 21 (2) (1996) 453–477. V. Kanovei, M. Reeken, Nonstandard Analysis: Axiomatically. Springer Monographs in Mathematics, 2004, XVI+408 p. V. Kanovei, V. Lyubetsky, Problems of set theoretic nonstandard analysis, Russ. Math. Surv. 62 (1) (2007) 45–111. G. Lolli, Infinitesimals and infinities in the history of mathematics, a brief survey, Appl. Math. Comput. 218 (2012) 7979–7988. Ya. D. Sergeyev, Arithmetic of infinity, Edizioni Orizzonti Meridionali, 2003, VIII+104 pp. Ya. D. Sergeyev, Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains, Non-linear Anal. Ser. A Theory Methods Appl. 71 (12) (2009) e1688–e1707. [10] Ya. D. Sergeyev, On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function, p-Adic Numbers, Ultrametric Anal. Appl. 3 (2) (2011) 129–148. [11] A. Zhigljavsky, Computing sums of conditionally convergent and divergent series using the concept of grossone, Appl. Math. Comput. 218 (2012) 8064– 8076.
