First return probabilities of birth and death chains and associated orthogonal polynomials

Abstract

Abstract. For a birth and death chain on the nonnegative integers, integral representations for first return probabilities are derived. While the integral representations for ordinary transition probabilities given by Karlin and McGregor (1959) involve a system of random walk polynomials and the corresponding measure of orthogonality, the formulas for the first return probabilities are based on the corresponding systems of associated orthogonal polynomials. Moreover, while the moments of the measure corresponding to the random walk polynomials give the ordinary return probabilities to the origin, the moments of the measure corresponding to the associated polynomials give the first return probabilities to the origin.As a by-product we obtain a new characterization in terms of canonical moments for the measure of orthogonality corresponding to the first associated orthogonal polynomials. The results are illustrated by several examples.

Random walk with long-range interaction with a barrier and its dual: Exact results

“…Proof: The only thing that remains to be proved is (6). The evaluation of π 0 can easily be derived from the Gauss identity involving two Gauss hypergeometric functions 2 F 1 .…”

“…Let τ 0,0 stand for the first return time to the origin of the RW starting from X 0 = 0. Applying the results in [6], we get an exact and asymptotic expression of the law of τ 0,0 :…”

“…Assume δ = 1, 3, 5, ... It was shown in (see [6], Theorem 2.1 and [7]) that the spectral measure of the first associated polynomials of a RW is related to the spectral probability measure of the dual RW through…”

“…As for the dual RW, we are then in the position to perform the singularity analysis of g even 1,0 (z) . The question of evaluating the large n scaling behavior of E 0 (X n ) only makes sense when δ ∈ (−1, 2] because when δ > 2, the RW is positive recurrent and has finite mean in the limit: E 0 (X n ) → E (X ∞ ) = x xπ x , with (π x ; x ≥ 0) the invariant probability measure of the chain given by (6) and (7).…”

Μελέτη των ριζών των associated ορθογωνίων q-πολυωνύμων

“…Ta associated orjog¸nia polu¸numa ektìc apì ta majhmatik probl mata sta opoÐa emfanÐzontai (bl. [19,99]) emfanÐzontai kai se poll probl mata thc fusik c. 'Etsi emfanÐzontai se probl mata thc kbantomhqanik c [102,112,113], se diadikasÐec gènnhshc kai jantou [21,66], k.t.l.. EpÐshc oi rÐzec touc brÐskoun efarmog sthn kataskeu diadikasi¸n bèltisthc parembol c (optimal interpolation processes) [88].…”