On singular univariate specializations of bivariate hypergeometric functions

It is tempting to evaluate F2(x,1) and similar univariate specializations of Appell's functions by evaluating the apparent power series at x=0 straight away using the Gauss formula for 2F1(1). But this kind of naive evaluation can lead to errors as the 2F1(1) coefficients might eventually diverge; then the actual power series at x=0 might involve branching terms. This paper demonstrates these complications on concrete examples.Comment: 10 page

Askey-Wilson relations and Leonard pairs

It is known that if $(A,B)$ is a Leonard pair, then the linear transformations $A$, $B$ satisfy the Askey-Wilson relations A^2 B - b A B A + B A^2 - g (A B+B A) - r B = h A^2 + w A + e I, B^2 A - b B A B + A B^2 - h (A B+B A) - s A = g B^2 + w B + f I, for some scalars $b,g,h,r,s,w,e,f$. The problem of this paper is the following: given a pair of Askey-Wilson relations as above, how many Leonard pairs are there that satisfy those relations? It turns out that the answer is 5 in general. We give the generic number of Leonard pairs for each Askey-Wilson type of Askey-Wilson relations.Comment: 22 pages; corrected version; the example of Section 2 has the normalization consistent with the rest of the pape

Contiguous relations of hypergeometric series

The 15 Gauss contiguous relations for ${}_2F_1$ hypergeometric series imply that any three ${}_2F_1$ series whose corresponding parameters differ by integers are linearly related (over the field of rational functions in the parameters). We prove several properties of coefficients of these general contiguous relations, and use the results to propose effective ways to compute contiguous relations. We also discuss contiguous relations of generalized and basic hypergeometric functions, and several applications of them.Comment: 12 pages; full bibliography added. This is the published text, with corrected formulas (24)-(25

Computation of highly ramified coverings

An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of these coverings with a fixed ramification pattern. (That is, Hurwitz spaces for these coverings are curves.) In this paper, three almost Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth Painleve equation.Comment: 26 page

Normalized Leonard pairs and Askey-Wilson relations

Let $V$ denote a vector space with finite positive dimension, and let $(A,B)$ denote a Leonard pair on $V$. As is known, the linear transformations $A,B$ satisfy the Askey-Wilson relations A^2B -bABA +BA^2 -g(AB+BA) -rB = hA^2 +wA +eI, B^2A -bBAB +AB^2 -h(AB+BA) -sA = gB^2 +wB +fI, for some scalars $b,g,h,r,s,w,e,f$. The scalar sequence is unique if the dimension of $V$ is at least 4. If $c,c*,t,t*$ are scalars and $t,t*$ are not zero, then $(tA+c,t*B+c*)$ is a Leonard pair on $V$ as well. These affine transformations can be used to bring the Leonard pair or its Askey-Wilson relations into a convenient form. This paper presents convenient normalizations of Leonard pairs by the affine transformations, and exhibits explicit Askey-Wilson relations satisfied by them.Comment: 22 pages; corrected version, with improved presentation of Section

Identities between Appell's and hypergeometric functions

Univariate specializations of Appell's hypergeometric functions F1, F2, F3, F4 satisfy ordinary Fuchsian equations of order at most 4. In special cases, these differential equations are of order 2, and could be simple (pullback) transformations of Euler's differential equation for the Gauss hypergeometric function. The paper classifies these cases, and presents corresponding relations between univariate specializations of Appell's functions and univariate hypergeometric functions. The computational aspect and interesting identities are discussed.Comment: 30 pages; A few wrong evaluations of the published version are rectified (see arXiv:0906.1861

Vaikų ir suaugusiųjų stuburo deformacijų chirurgijos ir anestezijos protokolas: priešoperacinio ir perioperacinio ištyrimo bei gydymo algoritmas, rekomenduojamas stuburo deformacijų chirurgijoje

Spinal deformity surgery is one of the most challenging surgeries that is only performed by highly professional multidisciplinary team in dedicated spinal centres. In the paper, the authors share and present the algorithm for safe and successful management of complex spinal disorders in Vilnius University Hospital Santaros Clinics.Stuburo deformacijų chirurgija – viena iš sudėtingiausių chirurgijos sričių. Operacijas gali atlikti tik profesionalios daugiaprofilinės specialistų komandos specializuotuose stuburo chirurgijos centruose.Straipsnyje pristatomas algoritmas, taikomas Vilnius universiteto ligoninės Santaros klinikų pacientams, kuriems diagnozuojama sudėtinga stuburo patologija, kai reikia stuburo rekonstrukcinės operacijos

The sixth Painleve transcendent and uniformization of algebraic curves

We exhibit a remarkable connection between sixth equation of Painleve list and infinite families of explicitly uniformizable algebraic curves. Fuchsian equations, congruences for group transformations, differential calculus of functions and differentials on corresponding Riemann surfaces, Abelian integrals, analytic connections (generalizations of Chazy's equations), and other attributes of uniformization can be obtained for these curves. As byproducts of the theory, we establish relations between Picard-Hitchin's curves, hyperelliptic curves, punctured tori, Heun's equations, and the famous differential equation which Apery used to prove the irrationality of Riemann's zeta(3).Comment: Final version. Numerous improvements; English, 49 pages, 1 table, no figures, LaTe

Towards all-order Laurent expansion of generalized hypergeometric functions around rational values of parameters

We prove the following theorems: 1) The Laurent expansions in epsilon of the Gauss hypergeometric functions 2F1(I_1+a*epsilon, I_2+b*epsilon; I_3+p/q + c epsilon; z), 2F1(I_1+p/q+a*epsilon, I_2+p/q+b*epsilon; I_3+ p/q+c*epsilon;z), 2F1(I_1+p/q+a*epsilon, I_2+b*epsilon; I_3+p/q+c*epsilon;z), where I_1,I_2,I_3,p,q are arbitrary integers, a,b,c are arbitrary numbers and epsilon is an infinitesimal parameter, are expressible in terms of multiple polylogarithms of q-roots of unity with coefficients that are ratios of polynomials; 2) The Laurent expansion of the Gauss hypergeometric function 2F1(I_1+p/q+a*epsilon, I_2+b*epsilon; I_3+c*epsilon;z) is expressible in terms of multiple polylogarithms of q-roots of unity times powers of logarithm with coefficients that are ratios of polynomials; 3) The multiple inverse rational sums (see Eq. (2)) and the multiple rational sums (see Eq. (3)) are expressible in terms of multiple polylogarithms; 4) The generalized hypergeometric functions (see Eq. (4)) are expressible in terms of multiple polylogarithms with coefficients that are ratios of polynomials.Comment: 48 pages in LaTe