## View abstract

### Session S32 - Special functions and orthogonal polynomials

Tuesday, July 13, 20:00 ~ 21:00 UTC-3

## Stochastic factorizations of birth-death chains and Darboux transformations

### Manuel D de la Iglesia

#### Universidad Nacional Autónoma de México, México   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak722080bc6faa90dca96514450ede61a2').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy722080bc6faa90dca96514450ede61a2 = 'md&#105;29' + '&#64;'; addy722080bc6faa90dca96514450ede61a2 = addy722080bc6faa90dca96514450ede61a2 + '&#105;m' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx'; var addy_text722080bc6faa90dca96514450ede61a2 = 'md&#105;29' + '&#64;' + '&#105;m' + '&#46;' + '&#117;n&#97;m' + '&#46;' + 'mx';document.getElementById('cloak722080bc6faa90dca96514450ede61a2').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy722080bc6faa90dca96514450ede61a2 + '\'>'+addy_text722080bc6faa90dca96514450ede61a2+'<\/a>';

Let $P$ be the transition probability matrix of a discrete-time birth-death chain, i.e. a stochastic Jacobi matrix. We consider factorizations of the form $P=P_1P_2$, where $P_1$ and $P_2$ are also stochastic matrices. By inverting the order of multiplication (also known as a Darboux transformation) we obtain new discrete-time birth-death chains $\widetilde{P}=P_2P_1$ from which we can identify the spectral measures associated with $\widetilde{P}$ from the original spectral measure and the corresponding orthogonal polynomials. We show several situations for different state spaces.

Joint work with F. A. Grünbaum (University of California, Berkeley) and C. Juarez (Universidad Nacional Autónoma de México).