# An Introduction to Ultrametric Summability Theory by P.N. Natarajan PDF

By P.N. Natarajan

ISBN-10: 8132225589

ISBN-13: 9788132225584

ISBN-10: 8132225597

ISBN-13: 9788132225591

This is the second one, thoroughly revised and extended variation of the author’s first ebook, protecting a number of new themes and up to date advancements in ultrametric summability concept. Ultrametric research has emerged as a huge department of arithmetic lately. This e-book offers a short survey of the study thus far in ultrametric summability idea, that is a fusion of a classical department of arithmetic (summability conception) with a latest department of study (ultrametric analysis). a number of mathematicians have contributed to summability idea in addition to useful research. The booklet will attract either younger researchers and more matured mathematicians who're trying to discover new parts in research. The e-book can be precious as a textual content in the event you desire to focus on ultrametric summability theory.

**Read or Download An Introduction to Ultrametric Summability Theory PDF**

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**Additional info for An Introduction to Ultrametric Summability Theory**

**Example text**

I=1 Proof Let A = (ank ) be a regular matrix. We can now choose two sequences of positive integers {n(m)} and {k(m)} such that if 36 4 Ultrametric Summability Theory m = 2 p, n(m) > n(m − 1), k(m) > k(m − 1) + (2m − 5)r, then k(m−1)+(2m−5)r |an(m),k | < 1 , 16 |an(m),k | < 1 ; 16 k=0 ∞ k=k(m−1) and if m = 2 p + 1, n(m) > n(m − 1), k(m) > k(m − 1) + (m − 2)r, then k(m−1)+(m−2)r |an(m),k | < k=0 1 , 16 k(m) |an(m),k | > k=k(m−1)+(m−2)r +1 ∞ |an(m),k | < k=k(m)+1 7 , 8 1 . 16 Define the sequence x = {xk } as follows: if k(2 p − 1) < k ≤ k(2 p), then xk = ⎧ 2 p−2 ⎪ ⎪ 2 p−1 , ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎪ 2 p−3 ⎪ ⎪ ⎪ 2 p−1 , ⎪ ⎪ ⎪ ⎪ 1, ⎪ ⎪ ⎪ ..

3 In Q3 , find the rational number whose canonical expansion is 2, 0121. As in the classical set up, in Q p too, we have the “exponential” and “logarithmic” series respectively defined by ∞ E(x) = n=0 xn and L(1 + x) = n! ∞ (−1)n−1 n=1 xn , n which converge for all x ∈ Q p with |x| p < 1. These series have properties which are very similar to their classical counterparts, say, for instance, E(x + y) = E(x)E(y); L((1 + x)(1 + y)) = L(1 + x) + L(1 + y); L(E(x)) = x; E(L(1 + x)) = 1 + x. In Q p , we have Binomial series too (for details, refer to [1]).

Xk = .. ⎪ ⎪ ⎪ 2 p−1 ⎪ , ⎪ ⎪ ⎪ 2p ⎩ 1, k(2 p) < k ≤ k(2 p) + r, k(2 p) + r < k ≤ k(2 p) + 2r, k(2 p) + (2 p − 2)r < k ≤ k(2 p) + (2 p − 1)r, k(2 p) + (2 p − 1)r < k ≤ k(2 p + 1). We note that, if k(2 p − 1) < k ≤ k(2 p), |xk+r − xk | < 1 , 2p − 1 while, if k(2 p) < k ≤ k(2 p + 1), |xk+r − xk | < 1 . 2p Thus |xk+r − xk | → 0, k → ∞, showing that x = {xk } ∈ However, |xk+1 − xk | = r. 2p − 2 , if k = k(2 p − 1) + (2 p − 3)r, p = 1, 2, . . 2p − 1 Hence, |xk+1 − xk | → 0, k → ∞ and consequently x ∈ / 1.

### An Introduction to Ultrametric Summability Theory by P.N. Natarajan

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