Formal Power Series : Number Theory #Formal #Power #Series #: #Number #Theory
In calculus an infinite series of the form 𝜮 a(n) = a(0) + a(1) x + a(2) x² +...+ a(n) x +... is called a power series in x.
Here both x and a(n) are real or complex numbers.
To each power series there corresponds a radius of convergence r > 0 such that the series converges absolutely if | x |<r and diverges if | x | > r.
Note:
Here the radius r can be +∞
Here in this, we consider power series from a different point of view. We call these power series as FORMAL power series to distinguish them from the ordinary power series of calculus.
In the formal power series, x is never assigned a numerical value.
In power series 𝜮 c(n) x", the symbol x" is simply a device for locating the position of the nth coefficient a(n).
The coefficient a(0) is called the constant coefficient of the series.
Let A(x)=a(n) x"; B(x) = b(n) x. Then
1. A(x)+B(x) iff a(n) = b(n) for all n > 0
2. A(x)+B(x)=(a(x)+b(x)) x".
3. A(x) B(x) = c(n) x where c(n) = a(k) b(n-k)
Here the sequence {c(n)} is called the Cauchy product of the sequences {a(n)} and {b(n)}.
4. The two operators sum and product of power series satisfy the associative, commutative laws together with the multiplication is distributive with respect to addition
i.e. for any power series A(x), B(x) and C(x),
i. [A(x)+B(x)] + C(x) = A(x) + [B(x) + C(x)] and [A(x), B(x)]. C(x) = A(x). [B(x), C(x)]
ii. A(x) + B(x) = B(x) + A(x) and A(x), B(x) = B(x). A(x) lii. A(x). [B(x) + C(x)] = A(x)B(x)+A(x)C(x)
Hence from the above algebraic properties, the set of formal power series Forms a ring with the zero element 0 = a(n) * x ^ n where a(n) = 0 for all n > 0 and the identity element 1 = a * (n) * x ^ n where a(0) = 1 and a(n) = 0 for all n > 1
Formal polynomial :
A formal power series
is called a formal polynomial if all its coefficients are 0 from some point on.
For each formal power series A(x) = 𝜮 a(n) xn
with non zero constant coefficient a(0) there is a
Uniquely determined formal power series B(x) = 𝜮 b(n) xn such that A(x)B(x) =1. Its coefficients
Can be determined by solving the infinite system of equations
a(0)b(0) = 1
a(0)b(1) +
a(1)b(0) = 0
a(0)b(2) +
a(1)b(1) + a(2)b(0) = 0
is succession for
b(0) , b(1) , b(2) , …. The series B(x) is called the inverse of A(x) and is
denoted by
A(x)-1 or by 1/A(x).
The special series A(x) = 1 + 𝜮 an xn is called a geometric series. Here a is an arbitrary real or Complex number. Its inverse is the formal polynomial
B(x) = 1 – ax.
In other words , we have 1/ (1-ax)
= 1 + 𝜮 an xn
* Fundamental Theorem of Arithmetic
* Historical Introduction to Number Theory
* The Mobius Function 𝝻 ( n ) .
#Formal #Power #Series #: #Number #Theory
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