Generalized Relation between the Roots of Polynomial and Term of Recurrence Relation Sequence

Abstract

Many researchers have been working on recurrence relation which is an important topic not only in mathematics but also in physics, economics and various applications in computer science. There are many useful results on recurrence relation sequence but there main problem to find any term of recurrence relation sequence we need to find all previous terms of recurrence relation sequence. There were many important theorems obtained on recurrence relations. In this paper we have given special identity for generalized kth order recurrence relation. These identities are very useful for finding any term of any order of recurrence relation sequence. Authors define a special formula in this paper by this we can find direct any term of a recurrence relation sequence. In this recurrence relation sequence to find any terms we need to find all previous terms so this result is very important. There is important property of a relation between coefficients of recurrence relation terms and roots of a polynomial for second order relation but in this paper, we gave this same property of recurrence relation of all higher order recurrence relation. So finally, we can say that this theorem is valid all order of recurrence relation only condition that roots are distinct. So, we can say that this paper is generalization of property of a relation between coefficients of recurrence relation terms and roots of a polynomial. Theorem: - Let C₁ and C₂ are arbitrary real numbers and suppose the equation <img src=image/13422456_01.gif> (1) Has X₁ and X₂ are distinct roots. Then the sequence <img src=image/13422456_25.gif> is a solution of the recurrence relation <img src=image/13422456_02.gif> (2) <img src=image/13422456_03.gif>. For n= 0, 1, 2 …where β₁ and β₂ are arbitrary constants. Proof: - First suppose that <img src=image/13422456_25.gif> of type <img src=image/13422456_04.gif> we shall prove <img src=image/13422456_25.gif> is a solution of recurrence relation (2). Since X₁, X₂ and X₃ are roots of equation (1) so all are satisfied equation (1) so we have<img src=image/13422456_05.gif>, <img src=image/13422456_06.gif>. Consider <img src=image/13422456_07.gif><img src=image/13422456_08.gif>. This implies <img src=image/13422456_09.gif>. So the sequence <img src=image/13422456_25.gif> is a solution of the recurrence relation. Now we will prove the second part of theorem. Let <img src=image/13422456_10.gif> is a sequence with three <img src=image/13422456_11.gif>. Let <img src=image/13422456_12.gif>. So <img src=image/13422456_13.gif> (3). <img src=image/13422456_14.gif> (4). Multiply by X₁ to (3) and subtracts from (4). We have <img src=image/13422456_15.gif> similarly we can find <img src=image/13422456_16.gif>. So we can say that values of β₁ and β₂ are defined as roots are distinct. So non- trivial values ofβ₁ and β₂ can find and we can say that result is valid. Example: Let <img src=image/13422456_25.gif> be any sequence such that <img src=image/13422456_17.gif> n≥3 and a₀=0, a₁=1, a₂=2. Then find a₁₀ for above sequence. Solution: The polynomial of above sequence is <img src=image/13422456_18.gif>. Solving this equation we have roots are 1, 2, and 3 using above theorem we have <img src=image/13422456_19.gif> (7). Using a₀=0, a₁=1, a₂=2 in (7) we have β₁+β₂+β₃=0 (8). β₁+2β₂+3β₂=1 (9).β₁+4β₂+9β₃=2 (10) Solving (8), (9) and (10) we have <img src=image/13422456_20.gif>, <img src=image/13422456_21.gif>, <img src=image/13422456_22.gif>. This implies <img src=image/13422456_23.gif>. Now put n=10 we have a₁₀=-27478. Recurrence relation is a very useful topic of mathematics, many problems of real life may be solved by recurrence relations, but in recurrence relation there is a major difficulty in the recurrence relation. If we want to find 100th term of sequence, then we need to find all previous 99 terms of given sequence, then we can get 100th term of sequence but above theorem is very useful if coefficients of recurrence relation of given sequence satisfies the condition of the above theorem, then we can apply above theorem and we can find direct any term of sequence without finding all previous terms.