Sequence conversion

Transformation of sequences - an operator acting on the space of sequences . The transformation of sequences includes such concepts as the convolution of one sequence with another, their summation and binomial transforms , as well as the Mobius and Stringing transforms . Sequence transformations can be used to accelerate the convergence of a series.

Content

1 Definition
2 Examples
3 Literature
4 References

Definition

Let a sequence be given $S=\{s_{n}\}_{n\in \mathbb {N} }.$ ${\ displaystyle S = \ {s_ {n} \} _ {n \ in \ mathbb {N}}.}$ Its transformation is denoted by $\mathbf {T} (S)=S'=\{s'_{n}\}_{n\in \mathbb {N} },$ ${\ displaystyle \ mathbf {T} (S) = S '= \ {s' _ {n} \} _ {n \ in \ mathbb {N}},}$ Where

s_{n}'=T(s_{n},s_{n+1},\dots ,s_{n+k}),

{\ displaystyle s_ {n} '= T (s_ {n}, s_ {n + 1}, \ dots, s_ {n + k}),}

and

s_{n}

{\ displaystyle s_ {n}}

, and

s'_{n}

{\ displaystyle s' _ {n}}

are either real or complex numbers . You can also generally consider them as elements of a vector space .

Converted Sequence $s'_{n}$ ${\ displaystyle s' _ {n}}$ converges faster than $s_{n}$ ${\ displaystyle s_ {n}}$ , if

\lim _{n\to \infty }{\frac {s'_{n}-\ell }{s_{n}-\ell }}=0,

{\ displaystyle \ lim _ {n \ to \ infty} {\ frac {s' _ {n} - \ ell} {s_ {n} - \ ell}} = 0,}

Where

\ell

{\ displaystyle \ ell}

- limit of convergent sequence

S

{\ displaystyle S}

.

If the mapping $T(s)$ ${\ displaystyle T (s)}$ linearly for each of its arguments, that is, if

s'_{n}=\sum _{m=0}^{k}c_{m}s_{n+m},

{\ displaystyle s' _ {n} = \ sum _ {m = 0} ^ {k} c_ {m} s_ {n + m},}

for some constants

c_{0},\cdots ,c_{k}

{\ displaystyle c_ {0}, \ cdots, c_ {k}}

then the transformation

T(s)

{\ displaystyle T (s)}

called a linear sequence transformation. If this condition is not met, then the transformation is called non-linear.

Examples

Binomial transformations ;
Mobius transformations ;
Shank transformations ;
Aitken's Delta-Square Transformation .

Literature

Hugh J. Hamilton, " Mertens' Theorem and Sequence Transformations ", AMS (1947)
Vorobiev N.N. Series Theory. - M .: Nauka, 1986 .-- 408 p.

Links

Transformations of Integer Sequences , a subpage of the On-Line Encyclopedia of Integer Sequences