Abstract
Generating function is a mathematical technique to concisely represent a known ordered sequence into a simple algebraic function. In essence, it takes a sequence as input, and produces a continuous function in one or more dummy (arbitrary) variables as output. A sequence is an ordered succession of elements which may be finite or infinite.
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Notes
- 1.
Analysis of algorithms is used to either select an algorithm with minimal run time from a set of possible choices, or minimum storage or network communication requirements. As the attribute of interest is time, the recurrences are developed as T(n) where n is the problem size. This is one of the reasons for choosing a dummy variable other than t.
- 2.
Note that the \(n^{th}\) derivative of \(1/(1-x)\) is \(n!/(1-x)^{n+1}\) and that of \(1/(1-ax)^b\) is \((n+b-1)! a^n/[(b-1)! (1-ax)^{b+n}]\).
- 3.
The sequence \(\sum _{n\ge 0}n!x^n\) converges only at \(x=0\).
- 4.
Some authors define it in the complex domain as \(\sum _{n=1}^\infty a_n [\exp (i\theta _n)]^{-s}\).
References
Basant, Z. P., & Wu, S. T. (1973). Dirichlet series creep function for aging concrete. Journal of the Engineering Mechanics Division, 99(2) (1973). https://doi.org/10.1061/JMCEA3.0001741.
Bona, M. (2012). Combinatorics of permutations (2nd edn.) CRC Press.
Boyadzhiev, K. N. (2016). Lah numbers, Laguerre polynomials of order negative one, and the n-th derivative of exp(1/x). Acta University Sapientiae, Mathematica, 8(1), 22–31. https://doi.org/10.1515/ausm-2016-0002.
Chattamvelli, R., & Jones, M. C. (1995). Recurrence relations for noncentral density, distribution functions, and inverse moments. Journal of Statistical Computation and Simulation, 52(3), 289–299. https://doi.org/10.1080/00949659508811679.
Daboul, S., et al. (2013). The Lah numbers and the n-th derivative of \(\exp (1/x)\). Mathematics Magazine, 86(1), 39–47.
Feiler, C., & Schleich, W. P. (2015). Dirichlet series as interfering probability amplitudes for quantum measurements. New Journal of Physics, 17, 063040. https://iopscience.iop.org/article/10.1088/1367-2630/17/6/063040.
Ghosal, S. K., Mukhopadhyay, S., Hossain, S., & Sarkar, R. (2020). Application of Lah transform for security and privacy of data through information hiding in telecommunication. In Transactions on Emerging Telecommunications Technologies (pp. 1–20). Wiley. https://doi.org/10.1002/ett.3984.
Hartleb, D., Ahrens, A., Purvinis, O., & Zascerinska, J. (2020). Analysis of free time intervals between buyers at cash register using generating functions. Proceedings of the 10th international conference on pervasive and parallel computing: Communication and sensors (PECCS2020) (pp. 42–49). ISBN: 978-989-758-4770.
Knuth, D. E. (1998). Fundamental algorithms (Vol. 2). Boston: Addison Wesley.
Lah, I. (1955). Eine neue art von zahlen, hire eigenschaftern and anwendung in Der mathematischen statistic. Mitteilungsblatt Mathematics Status, 7, 203–216.
Nkonkobe, S., & Murali, V. (2017). A study of a family of generating functions of Nelsen-Schmidt type and some identities on restricted barred preferential arrangements. Discrete Mathematics, 340(5), 1122–1128. https://doi.org/10.1016/j.disc.2016.11.010.
Riordan, J. (1979). Combinatorial identities. New York: Wiley.
Sedgewick, R., & Flajolet, P. (2013). An introduction to the analysis of algorithms. MA: Addison-Wesley.
Shishebor, Z., Nematollahi, A. R., & Soltani1, A. R. (2006). On covariance generating functions and spectral densities of periodically correlated autoregressive processes. Journal of Applied Mathematics and Stochastic Analysis, 2006, Article ID 94746. https://www.hindawi.com/journals/ijsa/2006/094746/ref/.
van Lint, J. H. (1999). Introduction to coding theory. Springer.
Zhang, J., Fan, R, & Shen, F. (2021). New method for the computation of generating functions with applications. In 2021 international conference on computational science and computational intelligence (CSCI). Las Vegas, NV, USA: IEEE. https://doi.org/10.1109/CSCI54926.2021.00156.
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Chattamvelli, R., Shanmugam, R. (2023). Types of Generating Functions. In: Generating Functions in Engineering and the Applied Sciences. Synthesis Lectures on Engineering, Science, and Technology. Springer, Cham. https://doi.org/10.1007/978-3-031-21143-0_1
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