Abstract. We introduce the quadratic harness condition and show that integrable quadratic harnesses have orthogonal martingale polynomials with a three step recurrence that satisfies a q-commutation relation. This implies that quadratic harnesses are essentially determined uniquely by five numerical constants. Explicit recurrences for the orthogonal martingale polynomials are derived in several cases of interest.
This paper is a continuation of our previous research on quadratic harnesses, that is, processes with linear regressions and quadratic conditional variances. Our main result is a construction of a Markov process from given orthogonal and martingale polynomials. The construction uses a two-parameter extension of the Al-Salam-Chihara polynomials and a relation between these polynomials for different values of parameters.
Highlights 1. Agreement between observers allows for statistical comparison of soil water repellency results 2. Best average CA compatibility was observed between the WDPT/MED and sessile drop methods 3. WDPT below 5 s relates to an average CA below 40° for hydrophilic samples 4. Good relationship between MED and CA were obtained for a range of hydrophobic samples 5. CA ranged from 110 to 130° for strongly hydrophobic samples (600 s < WDPT < 3600 s)
Abstract. We solve the connection coefficient problem between the Al-Salam-Chihara polynomials and the q-Hermite polynomials, and we use the resulting identity to answer a question from probability theory. We also derive the distribution of some Al-Salam-Chihara polynomials, and compute determinants of related Hankel matrices.
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