# Generating functions and their applications

2010
Bilgin, Begül
Generating functions are important tools that are used in many areas of mathematics and especially statistics. Besides analyzing the general structure of sequences and their asymptotic behavior; these functions, which can be roughly thought as the transformation of sequences into functions, are also used effciently to solve combinatorial problems. In this thesis, the effects of the transformations of generating functions on their corresponding sequences and the effects of the change in sequences on the generating functions are examined. With these knowledge, the generating functions for the resulting sequence of some combinatorial problems such as number of partitions, number of involutions, Fibonacci numbers and Catalan numbers are found. Moreover, some mathematical identities are proved by using generating functions. The sequences are the bases of especially symmetric key cryptosystems in cryptography. It is seen that by using generating functions, linear complexities and periods of sequences generated by constant coeffcient linear homogeneous recursions, which are used in linear feedback shift register (LFSR) based stream ciphers, can be calculated. Hence studying generating functions leads to have a better understanding in them. Therefore, besides combinatorial problems, such recursions are also examined and the results are used to observe the linear complexity and the period of LFSR’s combined in different ways to generate “better” system of stream cipher.

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Citation Formats
B. Bilgin, “Generating functions and their applications,” M.S. - Master of Science, Middle East Technical University, 2010. 