International Journal of Statistics and Applied Mathematics 2017; 2(6): 261-263

ISSN: 2456-1452 Maths 2017; 2(6): 261-263 © 2017 Stats & Maths A novel approach of determining Stella Octangula www.mathsjournal.com Received: 03-09-2017 number and using initial value theorem Accepted: 04-10-2017 in Z - Transform G Janaki Department of Mathematics, Cauvery College for Women, G Janaki and S Vidhya Trichy-18, Tamil Nadu, India

S Vidhya Abstract Department of Mathematics, In this communication, we evaluate Stella Octangula number and Pronic number by applying the initial Cauvery College for Women, value theorem in Z – Transform. Trichy-18, Tamil Nadu, India Keywords: Inverse Z – Transform, Pronic number, Stella Octangula number

Notations 2 SO  n 2 n  1 n = Stella Octangula number of rank n .

Pro  n n  1 n = Pronic number of rank n .

Introduction Life is full of patterns; especially Mathematics is also full of patterns. The main aim of is to discover interesting and unexpected relationships. In [1-7], theory of numbers was discussed. The Z – Transform is useful for the manipulation of discrete date sequences and has acquired a new significance in the formulation and analysis of discrete-time systems. It is used extensively today in the areas of applied mathematics, digital signal

processing, control theory, population science, and economics. These discrete models are solved with difference equations in a manner that is analogous to solving continuous models with differential equations. In [8-10], Z-Transform methods were analysed. Recently in [11], the sequence of m-gonal numbers and octahedral numbers was developed. In this communication, we evaluate Stella Octangula number and Pronic number by using

initial value theorem in Z-Transform.

Definition

u n  0,1, 2,  u  0 If the function n is defined for discrete values and n for n  0 , then

Z u  U ( z )  u z  n  n   n its Z – Transform is defined to be . The inverse Z – Transform  1 Z U ( z )   u n . is written as

Initial Value Theorem

u  Lt U ( z ) Z (u )  U ( Z ), 0 n z   If then .

Method of Analysis The process of finding the Stella Octangula number and Pronic number by using initial value Correspondence theorem in Z – Transform is given in the following theorems. S. Vidhya Department of Mathematics, Cauvery College for Women, Trichy-18, Tamil Nadu, India ~261~ International Journal of Statistics and Applied Mathematics

Theorem 1:  z 3  10 z 2  z  Z 1  SO  4  n  z  1  

Proof Assume that z 3  10 z 2  z U ( z )  . 4  z  1

By initial value theorem, we have

u 0  Lt U ( z )  0 z     z 3  10 z 2  z   u  Lt z U ( z )  u   Lt z    0  1 1 0   4   z   z   ( z  1)      z 3  10 z 2  z 1  u  Lt z 2 U ( z )  u  u z 1  Lt z 2   14  2 2  2 2  1 2   0 1   4    z   z   ( z  1) z    z 3  10 z 2  z 1 14  u  Lt z 3 U ( z )  u  u z 1  u z  2  Lt z 3    51  3 2  3 2  1 3   0 1 2   4 2    z   z   ( z  1) z z    z 3  10 z 2  z 1 14 51  u  Lt z 4 U ( z )  u  u z 1  u z  2  u z  3  Lt z 4     124  4 2  4 2  1 4   0 1 2 3   4 2 3    z   z    ( z  1) z z z  5  1  2  3  4 u 5  Lt z U ( z )  u 0  u 1 z  u 2 z  u 3 z  u 4 z  z  

 z 3  10 z 2  z 1 14 51 124   Lt z 5      245  5 2  5 2  1  4 2 3 4    z    z  1 z z z z  

Continuing the above process, we get n  1  2  3  ( n 1) u n  Lt z U ( z )  u 0  u 1 z  u 2 z  u 3 z    u n 1 z  z  

 z 3  10 z 2  z 1 14 51   Lt z n       4 2 3  z    z  1 z z z   u  n 2 n 2  1  SO n n Thus,

Theorem 2:  2 z 2  Z 1  Pro  3  n  z  1  

Proof Assume that 2 z 2 U ( z )  . 3  z  1

By initial value theorem, we have

  2 z 2   u  Lt z U ( z )  u   Lt z    0  2  11  1 1 0   3   z   z   ( z  1)     ~262~ International Journal of Statistics and Applied Mathematics

 2 z 2 2  u  Lt z 2 U ( z )  u  u z 1  Lt z 2   6  2 2  1 2   0 1   3  z   z   ( z  1) z    2 z 2 2 6  u  Lt z 3 U ( z )  u  u z 1  u z  2  Lt z 3    12  33  1 3   0 1 2   3 2  z   z   ( z  1) z z    2 z 2 2 6 12  u  Lt z 4 U ( z )  u  u z 1  u z  2  u z  3  Lt z 4     20  4 4  1 4   0 1 2 3   3 2 3  z   z    ( z  1) z z z  5  1  2  3  4 u 5  Lt z U ( z )  u 0  u 1 z  u 2 z  u 3 z  u 4 z  z  

 2 z 2 2 6 12 20   Lt z 5      30  5 5  1  3 2 3 4  z    z  1 z z z z  

Continuing the above process, we get n  1  2  3  ( n 1) u n  Lt z U ( z )  u 0  u 1 z  u 2 z  u 3 z    u n 1 z  z  

 2 z 2 2 6 12   Lt z n       3 2 3  z    z  1 z z z   u  n n  1  Pro n n Thus,  2 z 2  Z 1  Pro  3  n  z  1  

Conclusion In this communication, we find Stella Octangula number and Pronic number by applying the initial value theorem in Z – Transform. To conclude that, one can find the other special numbers by using various properties of Z – Transform.

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