A note on the Neyman–Rayner triangle

A. G. Shannon
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 27, 2021, Number 4, Pages 164—166
DOI: 10.7546/nntdm.2021.27.4.164-166
Download full paper: PDF, 72 Kb


Authors and affiliations

A. G. Shannon
Warrane College, The University of New South Wales
Kensington, NSW, 2033, Australia


This note raises questions for other number theorists to tackle. It considers a triangle arising from some statistical research of John Rayner and his use of some orthonormal polynomials related to the Legendre polynomials. These are expressed in a way that challenges the generalizing them. In particular, the coefficients are expressed in a triangle and related to known sequences in the Online Encyclopedia of Integer Sequences. The note actually raises more questions than it answers when it links with the cluster algebra of Fomin and Zelevinsky.


  • Neyman–Rayner triangle
  • Orthonormal polynomials
  • Legendre polynomials
  • Cluster algebra

2020 Mathematics Subject Classification

  • 11S05
  • 11C08
  • 11N30


  1. Al-Salam, W. (1961). A generalized Turán expression for the Bessel functions. The American Mathematical Monthly, 68(2), 146–149.
  2. Bera, A. K., & Ghosh, A. (2019). Neyman’s Smooth Test and its Applications in Econometrics. In A. Ullah, A. T. K. Wan & A. Chaturverdi (Ed.), Handbook of Applied Econometrics and Statistical Inference. (pp. 177–230). Baton Rouge, FL: CRC Press.
  3. Dattoli, G., Migliorati, M., & Srivastava, H. M. (2004). Some families of generating functions for the Bessel and related functions. Georgian Mathematical Journal, 11(2), 219–228.
  4. Fomin, S. & Zelevinsky, A. (2002). Cluster algebras I: Foundations. Journal of the American Mathematical Society, 15(2), 497–529.
  5. Neyman, J. (1937). ‘Smooth’ test for goodness of fit. Scandinavian Actuarial Journal, 3–4, 149–199.
  6. Plackett, R. L. (1983). Karl Pearson and the Chi-Squared Test. International Statistical Review, 51(1), 59–72.
  7. Rayner, J. W. C., & Best, D. J. (1986). Neyman-type Smooth Tests for Location-Scale Families. Biometrika, 73(2), 437–446.
  8. Sloane, N. J. A. (1964). The On-Line Encyclopedia of Integer Sequences. Retrieved from: https://oeis.org.

Related papers

Cite this paper

Shannon, A. G. (2021). A note on the Neyman–Rayner triangle. Notes on Number Theory and Discrete Mathematics, 27(4), 164-166, doi: 10.7546/nntdm.2021.27.4.164-166.

Comments are closed.