Abstract
This paper considers the representation of odd moments of the distribution of a four-step uniform random walk in even dimensions, which are based on both linear combinations of two constants representable as contiguous very well-poised generalized hypergeometric series and as even moments of the square of the complete elliptic integral of the first kind. Neither constants are currently available in closed form. New symmetries are found in the critical values of the L-series of two underlying cusp forms, providing a sense in which one of the constants has a formal counterpart. The significant roles this constant and its counterpart play in multidisciplinary contexts is described. The results unblock the problem of representing them in terms of lower-order generalized hypergeometric series, offering progress towards identifying their closed forms. The same approach facilitates a canonical characterization of the hypergeometry of the parbelos, adding to the characterizations outlined by Campbell, D'Aurozio and Sondow (2020, The American Mathematical Monthly127(1), 23-32). The paper also connects the econometric problem of characterizing the bias in the canonical autoregressive model under the unit root hypothesis to very well-poised generalized hypergeometric series. The confluence of ideas presented reflects a multidisciplinarity that accords with the approach and philosophy of Prasanta Chandra Mahalanobis.
Highlights
In his assessment of the impact of Karl Pearson's work in the development of Statistics in India, Nayak (2009) outlined the nascent role it played in attracting Prasanta Chandra Mahalanobis to the discipline, and in inspiringMOMENTS IN PEARSON'S FOUR-STEP...S245 him towards his eventual leading role in its development across the Indian subcontinent
In the context of constructing the hypergeometry of the parbelos, we show that finding transformed series with entries in and provides a route towards finding the closed form of a relevant generalized hypergeometric series
Like the moments in the four-step uniform random walk problem, the constant M is representable in terms of very well-poised (VWP) 7F6(1) series
Summary
In his assessment of the impact of Karl Pearson's work in the development of Statistics in India, Nayak (2009) outlined the nascent role it played in attracting Prasanta Chandra Mahalanobis to the discipline, and in inspiring. While closed forms are available for N = 1 and 2, the small-N behaviour of (1.3) makes finding closedforms or even just tractable expressions for the densities and moments in the shortwalk problems analytically and numerically challenging. This is especially so for N = 3, 4, 5 and 6. A connection is made between the econometric problem of characterizing the bias in the canonical autoregressive model under the unit root hypothesis and VWP generalized hypergeometric series This elucidates a shared characteristic with the four-step random walk problem that could be exploited in future work.
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