Values of Hypergeometric Functions

The investigation of the arithmetic nature of the values of differentiated with respect to parameter generalized hypergeometric functions was carried out in many works; see [1]–[7] and also corresponding chapters of the books [8] and [9]. Primarily the method of Siegel was used for these purposes. This method can be applied for the investigation of hypergeometric functions with rational parameters and the results concerning transcendence and algebraic independence of the values of such functions and corresponding quantitative results (for example estimates of the measures of algebraic independence) were obtained by means of it. The possibilities of application of Siegel’s method in case of hypergeometric functions with irrational parameters are restricted. In its classic form Siegel’s method cannot be applied in this situation and here were required some new considerations. But it must be noted that the most general results concerning the arithmetic nature of the values of hypergeometric functions with irrational parameters were obtained exactly by Siegel’s method (by modified form of it, see [10] and [11]). In this case it’s impossible to say of the results of transcendence or algebraic independence and one must restrict oneself by the results concerning linear independence of the corresponding values. In Siegel’s method reasoning begins with the construction of functional linear approximating form which has a sufficiently high order of zero at the origin of coordinates. Such a form is constructed by means of the Dirichlet principle. The impossibility to realize the corresponding reasoning for the functions with irrational parameters is an obstacle for the attempts to apply Siegel’s method in case of irrational parameters. It was noted long ago that in some cases the linear approximating form can be constructed effectively and explicit formulae can be pointed out for its coefficients. This method is inferior to Siegel’s one in the sense of the generality of the results obtained. But by means of the method based on the effective construction of linear approximating form the most precise low estimates of the modules of linear forms in the values of hypergeometric functions were obtained and in many cases were established linear independence of the values of functions with irrational parameters (see for example [12]). The effective construction of linear approximating form for the function (2) was proposed in the work [13]. In this work the construction was based on a contour integral which was earlier used for the achievement of results concerning the estimates of linear forms of the values of hypergeometric functions with different parameters; see [14]. In this paper we propose a new approach for the construction of linear approximating form for functions (2). Here we make use of a connection between hypergeometric functions of different types which makes it possible to reduce above mentioned constructing of linear approximating form to less difficult task. In the conclusion we give short directions concerning possible applications.

Let 𝔽q be a finite field with q = pe elements, where p is a prime. We calculate the number of zeros of the hyperelliptic curve by using the properties of the quadratic character. With the help of these zeros we discuss the trace of Frobenius map on the hyperelliptic curves in terms of special values of 2F1 hypergeometric functions. Also, we find some identities, and particular values of hypergeometric functions over 𝔽q.

In this paper we consider some hypergeometric functions whose parameters are connected in a special way. Lower estimates of the moduli of linear forms in the values of such functions have been obtained. Usually for the achievement of such estimates one makes use of Siegel’s method; see [1], [2], [3, chapt. 3]. In this method the reasoning begins with the construction by means of Dirichlet principle of the linear approximating form having a sufficiently large order of zero at the origin of coordinates. Employing the system of differential equations, the functions under consideration satisfy, one constructs then a set of forms such that the determinant composed of the coefficients of the forms belonging to this set must not be equal to zero identically. Further steps consist of constructing a set of numerical forms and of proving of the interesting for the researcher assertions: linear independence of the values of the functions under consideration can be proved or corresponding quantitative results can be obtained. By means of Siegel’s method have been proved sufficiently general theorems concerning the arithmetic nature of the values of the generalized hypergeometric functions and in addition to aforementioned linear independence in many cases was established the transcendence and algebraic independence of the values of such functions. But the employment of Dirichlet principle at the first step of reasoning restricts the possibilities of the method. Its direct employment is possible in the case of hypergeometric functions with rational parameters only. It must be taken into consideration also the insufficient accuracy of the quantitative results that can be obtained by this method. As a consequence of these facts some analogue of Siegel’s method has been developed (see [4]) by means of which it became possible in some cases to investigate the arithmetic nature of the values of hypergeometric functions with irrational parameters also. But yet earlier one had begun to apply methods based on effective construction of linear approximating form. By means of such constructions the arithmetic nature of some classic constants was investigated and corresponding quantitative results were obtained, see for example [5, chapt. 1]. Subsequently it turned out that effective methods can be applied also for the investigation of generalized hypergeometric functions. Explicit formulae for the coefficients of the linear approximating forms were obtained. In some cases these formulae make it possible to realize Siegel method scheme also for the hypergeometric functions with irrational parameters. If in (1) polynomial a(x) is equal to unity identically then the results obtained by effective method are of sufficiently general nature and in this case further development of this method meets the obstacles of principal character. In case a(x) ≡ 1, however, the possibilities of effective method are not yet exhausted and the latest results can be generalized and improved. In the theorems proved in the present paper new qualitative and quantitative results are obtained for some hypergeometric functions with a(x) = x+α and polynomial b(x) from (1) of special character. The case of irrational parameters is under consideration but the ideas we use will apparently make it possible in the future to obtain new results in case of rational parameters also.

In the present paper we will consider the generalization of some methods for evaluation of irrationality measures for yd = pd lnppd+1 d1 and currently known results overview. The extent of irrationality for various values of Gauss hypergeometric function were estimated repeatedly, in particular for 2F(1; 1 2 ; 3 2 ; 1 d ) = p d ln p pd+1 d1 : The rst such estimates in some special cases were obtained by D. Rhinn [1], M. Huttner [2], D. Dubitskas [3]. Afterward by K. Vaananen, A. Heimonen and D. Matala-Aho [4] was elaborated the general method, which one made it possible to get upper bounds for irrationality measures of the Gauss hypergeometric function values F(1; 1k ; 1 + 1k ; rs ); k 2 N; k > 2; rs 2 Q; (r; s) = 1; r s 2 (1; 1): This method used the Jacobi type polynomials to construct rational approach to the hypergeometric function. In [4] have been obtained many certain estimates, and some of them have not been improved till now. But for the special classes of the values of hypergeometric function later were elaborated especial methods, which allowed to get better evaluations. In the papers [5], [6] authors, worked under supervision of V.Kh.Salikhov, obtained better estimates for the extent of irrationality for some specic values d: In the basis of proofs for that results were lying symmetrized integral constructions. It should be remarked, that lately symmetrized integrals uses very broadly for researching of irrationality measures. By using such integrals were obtained new estimates for ln 2( [7]),ln 3; ln , ( [8], [9]) and other values. Here we present research and compare some of such symmetrized constructions, which earlier allowed to improve upper bounds of irrationality measure for specic values of yd.

Let $X_0^6(1)/W_6$ be the Atkin-Lehner quotient of the Shimura curve $X_0^6(1)$ associated to a maximal order in an indefinite quaternion algebra of discriminant $6$ over $\mathbb Q$. By realizing modular forms on $X_0^6(1)/W_6$ in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer's formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over $\overline\mathbb Q$ with complex multiplication.

Let X 0 6 ( 1 ) / W 6 X_0^6(1)/W_6 be the Atkin–Lehner quotient of the Shimura curve X 0 6 ( 1 ) X_0^6(1) associated to a maximal order in an indefinite quaternion algebra of discriminant 6 6 over Q \mathbb {Q} . By realizing modular forms on X 0 6 ( 1 ) / W 6 X_0^6(1)/W_6 in two ways, one in terms of hypergeometric functions and the other in terms of Borcherds forms, and using Schofer’s formula for values of Borcherds forms at CM-points, we obtain special values of certain hypergeometric functions in terms of periods of elliptic curves over Q ¯ \overline {\mathbb {Q}} with complex multiplication.

Арифметические свойства значений гипергеометрической функции изучались различными методами, начиная с работы К.~Зигеля 1929 г.. Это направление теории диофантовых приближений исследовалось такими авторами как М.~Хата [1]-[2], Ф.~Аморозо и К.~Виола [3], А.~Хеймонен, В.~Матала-Ахо и К.~Ваананен [4]-[5] и многими другими. В последние десятилетия был получен ряд интересных результатов в этой области, усилено много ранее известных оценок меры иррациональности, как для значений гипергеометрической функции, так и для других величин.В настоящее время одним из широко применяемых подходов при построении оценок показателя иррациональности является использование интегральных конструкций, симметричных относительно какой-либо замены параметров. Симметризованные интегралы и ранее использовались разными авторами, например, в работе Дж.~Рина [6], но наиболее активное развитие это направление приобрело после работы В.~Х.~Салихова [7], получившего с помощью симметризованного интеграла новую оценку для $\ln{3}$. Впоследствии симметричность различного типа позволила доказать ряд значимых результатов. Были получены новые оценки для некоторых значений логарифмической функции, функции $\arctg{x}$, классических констант (см., например, [8] -- [18]). В 2014~г., используя общие симметризованные многочлены первой степени вида $At-B$, где $t=(x-d)^2$, К.~Ву и Л.~Ванг усилили результат В.~Х.~Салихова о мере иррациональности $\ln{3}$ (см.[19]). В работе [20] идея симметричности была применена к интегралу Р.~Марковеккио, доказавшего ранее новую оценку для $\ln{2}$ в [21], что позволило улучшить результат для $\pi/3$.Данная статья является продолжением работы [22], обобщающей результаты для двух типов симметричных интегральных конструкций. Первая позволяет более эффективно оценить показатели иррациональности чисел вида $\sqrt{d}\ln{\frac{\sqrt{d}+1}{\sqrt{d}-1}}$ при $d=2^{2k+1}, d=4k+1$ для некоторых $k\in\mathbb N$ (см. [22]). Используя данный интеграл, также можно получить оценки меры иррациональности чисел $\sqrt{4k+3}\ln{\frac{\sqrt{4k+3}+1}{\sqrt{4k+3}-1}},\ k\in\mathbb N$. Вторая рассматриваемая интегральная конструкция дает возможность оценивать меру иррациональности некоторых значений логарифмической функции, используя симметричность другого типа, что было подробно рассмотрено в [22]. Данный интеграл позволяет также оценивать меру иррациональности значений $\frac{1}{\sqrt{k}}\arctg{\frac{1}{\sqrt{k}}}$. Обобщение этого случая предлагается в данной работе.

In 1914, Ramanujan gave a list of 17 identities expressing $1/\pi$ as linear combinations of values of hypergeometric functions at certain rational numbers. Since then, identities of similar nature have been discovered by many authors. Nowadays, one of the standard approaches to this kind of identities uses the theory of modular curves. In this paper, we will consider the case of Shimura curves and obtain Ramanujan-type formulas involving special values of hypergeometric functions and products of Gamma values. The product of Gamma values are related to periods of elliptic curves with complex multiplication by Q(\sqrt{-3}) and Q(\sqrt{-4}).

The general problem of Pade approximation to a system of functions satisfying linear differential equations is considered. We use the method of isomonodromy deformation to construct effectively the remainder function and Pade approximants in the case of N-point approximations of solutions of Fuchsian linear differential equations. Special attention is devoted to generalized hypergeometric functions gFp (a1,…,a2;b1,…,bp;x). In various cases the asymptotics of the remainder function is presented. A separate section is devoted to applications of analytic methods to the problem of rational and diophantine approximations of the values at rational and algebraic points of functions satisfying linear differential equations. There is presented also an analytic method for investigation of the arithmetic nature of the constants, arising as values of hypergeometric functions, such as L2 (1/q): q ≥14, ζ(2), ζ(3) etc.. which occur in many physical situations.

Although numerical procedures often supply a required accuracy, closed-form expressions allow one to escape any accumulation of errors. In this paper, we discuss the possibility of obtaining explicit results for a variance gamma process. We derive the exact distribution of the maximum of the variance gamma process over a finite interval of time and establish the prices of path-dependent options including digital barrier, fixed-strike lookback, and lookback options. The obtained formulas are based on values of hypergeometric functions.

We derive analytic formulas for the prices of financial instruments in foreign currency within the framework of a stochastic model defined as the sum of a variance gamma and a Poisson process. We obtain our results for various types of dependencies in the model. The resulting formulas contain values of hypergeometric functions. Practical applications of our results include control over the activity of investors in financial markets.

The integer set previously obtained by the author in the study of moments and cumulants of three-parameter probability distribution of the hyperbolic cosine type is considered. This distribution is a generalization of Meixner two-parameter distribution. Moments of distribution at specific parameters vary as a certain class of polynomials with the corresponding coefficients. On the basis of the differential ratio of polynomials, recurrence formulas for their coefficients are received. The set of polynomial coefficients { U ( n ; k , j )} that depends on three indices, and which is formed by these formulas, is the object of study. The set is structured in the form of a numeric prism. When fixing one or two indices or functional connection between the indices, different sections of numerical prisms are obtained: number triangles or number sequences. Among the sections of the numerical prism are both known (Stirling triangle, tangential numbers, secant numbers, etc.) and new integer sets. Classic Bessel triangle enters into the considered numerical prism as a section { U (2 n–j ; n , j )}, where n = 0, 1, 2, …, j = 0, 1, 2, … n . In this section the sequences classified as coefficients in the Bessel polynomials are determined. Based on the theoretical developments related to the Bessel polynomials, dependences and relations for a number of elements of numerical prism are found and justified. The obtained results also allow putting sequences through the values of hypergeometric functions and modified Bessel functions of the second kind. Considered set differs in the ease of construction, and its study has revealed previously unknown properties and relations of various mathematical objects (sequences, polynomials, functions, etc.), particularly related to the Bessel polynomials

In this paper, estimates for nonhomogeneous linear forms in values of hypergeometric functions with irrational parameters are refined. This refinement is carried out by means of an optimal choice of the degree of the zeroth polynomial appearing in the construction of approximate functional forms.

<p>In this paper we consider a problem concerning the linear independence of the values of generalized hypergeometric functions with different irrational parameters. To solve this problem it is impossible to apply directly Siegel's method known in the theory of transcendental numbers, since abovementioned functions do not belong to a special class of entire functions Siegel has introduced. The method of Siegel uses a Dirichlet principle to construct linear approximating form. Such a construction cannot be realized if the coefficients of Taylor series of the functions under consideration have \\bad" denominators. We have exactly this circumstance in the case of hypergeometric functions with irrational parameters. There exists a modification of the method of Siegel, which allows us to apply the Dirichlet principle in this case also, but it is still unknown if this modification can be used in a situation where the varied parameters are irrational.</p><p>Usually in the case of irrational parameters one applies effective methods for constructing linear approximating form. If we apply such a construction in the case of different irrational parameters we shall have to confine ourselves by only two parameters and introduce the additional requirement: the difference of these parameters must be a rational number. The practice of considering similar problems for hypergeometric functions with irrational parameters shows that the use of simultaneous approximations leads as a rule to better results. In case of problems with different irrational parameters it is possible, for example, to waive the condition of rationality of the varied parameters difference.</p><p>In this paper, the effective construction of simultaneous approximations differs from previously proposed, which made it possible owing to some additional considerations of a technical nature to improve the arithmetic part of the method (which is, mainly, to attain an acceptable estimate of the least common denominator of the coefficients of the approximating polynomials). As a result the above mentioned unnecessary restriction on the varied parameters was dropped, but we have to restrict ourselves to the values of the functions only at the point with a small absolute value.</p><p>Remaining within the methods used in this paper one can obtain some other results which are not discussed in the article, however, increasing the number of varied parameters (if only to three) or rejecting the restriction on the values of the functions will require, probably, some new ideas.</p>

In this paper we primarily obtain the explicit formulas for the distribution function of the variance gamma process. The formulas are based on values of hypergeometric functions. This result is applied to European option pricing. Basing on the established formulas, we get the prices of binary options, as long as the price of European call which was derived firstly in paper by Madan, Carr and Chang (1998).