On the Use of Special Bilinear Functions in Computing Bernoulli Polynomials

In this paper we review, firstly, the subject of bilinear functions in connection with the convolution of two n-tuples vectors(originally named quacroms); then we summarize some of its important applications such as the calculation of the product of two polynomials and hence two integers and their use in representing certain quantities .The main topic of this paper is the use of these special bilinear functions in computing special functions as in the case of Bernoulli polynomials through simple recurrence relations ,this will be performed next and where the related algorithm will be described .Moreover, the first few Bernoulli polynomials are calculated. .


INTRODUCTION
Special bilinear functions in connection with the convolution of two n-tuples vectors a  and b  were introduced and studied ] 1 [ , the name " quacroms of dimension n 2 " was given to them then. The original applications for them were taking the product of two polynomials or of two integers and where the operation was shown to be more efficient and neater than the traditional way of doing that. More applications were found for them ,applications such as using them in representing certain quantities and their applications in solving linear equations which showed to be very useful as we will discuss in this paper(linear quacrom equations) ] 2 [ .Quacroms of the dimension n 3 were then introduced and discussed ] 3 [ .In the next section we give some details regarding these special bilinear functions(SBF) ;then we advance to show some important applications .In the section to follow ,we describe an algorithm to compute Bernoulli polynomials using SBF followed by practical calculations. Finally we conclude with a short discussion.

Definition 1
Consider a real-valued function , which has the following properties Then f is the SBF (or quacrom) of A written as

Some Properties and Remarks
In this subsection ,we present some properties which can be verified using definition 1 [1];  ,i.e. rows can be interchanged.
,which means that the scalar can be taken out as a common factor for rows; this is not the case for columns. with the binary operation "addition" does not form a group for a fixed n.
i-The notation for the SBF originally was ,but due to technical difficulties we replaced it by . Where we have to note that an integer N of n digits can be written as

1-If
We note here that this method of taking the product is different from the traditional one .It is easier, faster ,and takes place in one line .Moreover it is applicable to all bases.
In practice to calculate any SBF(quacrom) -put in an array form-we imagine that a pair of scissors is opened with angle with its two ends joining the first and the th n columns; we multiply crosswise and add, then we start closing the scissors repeating the process of crosswise multiplication whenever the ends meet with any digits until it is completely closed .This is where the word "quacrom" came from.
To clarify the above remarks we give the following example
Therefore the product is 68373 and we should note that performing the process in the manner we showed is very formal but in practice the process is very quick and the result is given in one line [1].   [2]. ;however this definition involving SBF will lead to a very important application and actually to an interesting algorithm which will enable us to compute Bernoulli polynomials in a simple and straitforward manner.

Bernoulli Polynomials as an Application
Bernoulli polynomials ) (x B n are generated by [5]  However, the numerator in the right hand side of Equation (7)   Comparing the numerator given by Equation (8) with the numerator in the left hand side of Equation(7), we get From Equation(9) we see that And in general for ,any i=n, we have We should note that Equation (12) In the following and implementing Equation (12) we proceed to describe our interesting algorithm which can be used to compute Bernoulli polynomials Step 1 Define a function of two variable vectors( an SBF) as in equation (1).
Step 2 Compute Bernoulli polynomials using the defined function and equations (5) and (12).The steps to get ) ( ) ( 2 0 x B x B are to be taken as a guide.
Step 3 Results are to be compared with the values given in Reference [4].

CONCLUSION
As we have seen SBF and LQE have many useful applications some of which are computing product of two polynomials and numbers,their use in expressing various quantities and finally their use in describing a method by which one can compute special functions such as Bernoulli polynomials.In fact one expect that such an algorithm can be used to calculate other polynomials as in the case of Legendre Polynomials.This will constitute the subject of a future study.