On the orthogonality of partial sums of the generalized hypergeometric series


UDC 517.587

It turns out that the partial sums $g_n(z)=\displaystyle\sum\nolimits_{k=0}^n\dfrac{(a_1)_k\ldots(a_p)_k}{(b_1)_k\ldots(b_q)_k}\,\dfrac{z^k}{k!}$ of the generalized hypergeometric series ${}_p F_q(a_1,\ldots,a_p;b_1,\ldots,b_q;z)$ with parameters $a_j, b_l\in\mathbb{C}\backslash\{0,-1,-2,\ldots\}$ are Sobolev orthogonal polynomials.
The corresponding monic polynomials $G_n(z)$ are $R_I$-type polynomials, and therefore, they are related to biorthogonal rational functions.
The polynomials $g_n$ satisfy a differential equation (in $z$) and a recurrence relation (in $n$).
In this paper, we study the integral representations for $g_n$ and their basic properties.
It is shown that partial sums of arbitrary power series with non-zero coefficients are also related to biorthogonal rational functions.
For polynomials $g_n(z),$ we obtain a relation to Jacobi-type pencils and their associated polynomials.


L. C. Andrews, Special functions of mathematics for engineers, Oxford Univ. Press, Oxford (1998).

E. Hendriksen, H. van Rossum, Orthogonal Laurent polynomials, Nederl. Akad. Wetensch. Indag. Math., 48, № 1, 17 – 36 (1986). DOI: https://doi.org/10.1016/1385-7258(86)90003-X

E. Horozov, $d$-orthogonal analogs of classical orthogonal polynomials, SIGMA Symmetry Integrability and Geom. Methods and Appl., 14, Articll 063 (2018), 27 p., https://doi.org/10.3842/SIGMA.2018.063 DOI: https://doi.org/10.3842/SIGMA.2018.063

E. Horozov, Vector orthogonal polynomials with Bochner’s property, Constr. Approx., 48, № 2, 201 – 234 (2018), https://doi.org/10.1007/s00365-017-9410-6 DOI: https://doi.org/10.1007/s00365-017-9410-6

M. E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia Math. and Appl., 98, Cambridge Univ. Press, Cambridge (2005), https://doi.org/10.1017/CBO9781107325982 DOI: https://doi.org/10.1017/CBO9781107325982

M. E. H. Ismail, D. R. Masson, Generalized orthogonality and continued fractions, J. Approx. Theory, 83, № 1, 1 – 40 (1995), https://doi.org/10.1006/jath.1995.1106 DOI: https://doi.org/10.1006/jath.1995.1106

F. Marcell´an, Yuan Xu, On Sobolev orthogonal polynomials, Expo. Math., 33, № 3, 308 – 352 (2015), https://doi.org/10.1016/j.exmath.2014.10.002 DOI: https://doi.org/10.1016/j.exmath.2014.10.002

M. Marden, Geometry of polynomials, Second ed., Math. Surveys and Monogr., № 3, Amer. Math. Soc., Providence, R.I. (1966).

E. D. Rainville, Special functions, First ed., Chelsea Publ. Co., Bronx, N.Y. (1971).

B. Simon, Orthogonal polynomials on the unit circle. Pt 1. Classical theory, Colloq. Publ., 54, Amer. Math. Soc., Providence, R.I. (2005), https://doi.org/10.1090/coll054.1 DOI: https://doi.org/10.1090/coll054.1

B. Simon, Orthogonal polynomials on the unit circle. Pt 2. Spectral theory, Colloq. Publ., 54, Amer. Math. Soc., Providence, R.I. (2005), https://doi.org/10.1090/coll/054.2/01 DOI: https://doi.org/10.1090/coll/054.2

V. Spiridonov, A. Zhedanov, Classical biorthogonal rational functions on elliptic grids, C. R. Math. Acad. Sci. Soc. R. Can., 22, № 2, 70 – 76 (2000).

G. Szeg¨o, Orthogonal polynomials, Fourth ed. Colloq. Publ., 23, Amer. Math. Soc., Providence, R.I. (1975).

S. M. Zagorodnyuk, Ortogonal`nye mnogochleny, assocy`y`rovannye s nekotorymy` puchkamy` yakoby`evogo ty`pa, Ukr. mat. zhurn., 68, № 9, 1180 – 1190 (2016).

S. M. Zagorodnyuk, On some classical type Sobolev orthogonal polynomials, J. Approx. Theory, 250, Article 105337 (2020), 14 p., https://doi.org/10.1016/j.jat.2019.105337 DOI: https://doi.org/10.1016/j.jat.2019.105337

S. M. Zagorodnyuk, On a family of hypergeometric Sobolev orthogonal polynomials on the unit circle, Constr. Math. Anal., 3, № 2, 84 – 75 (2020), https://doi.org/10.33205/cma.690236 DOI: https://doi.org/10.33205/cma.690236

S. M. Zagorodnyuk, On series of orthogonal polynomials and systems of classical type polynomials, Укр. мат. журн., 73, № 6, 799 – 810 (2021), https://doi.org/10.37863/umzh.v73i6.6527 DOI: https://doi.org/10.1007/s11253-021-01968-1

A. Zhedanov, The ”classical” Laurent biorthogonal polynomials, J. Comput. and Appl. Math., 98, № 1, 121 – 147 (1998), https://doi.org/10.1016/S0377-0427(98)00118-6 DOI: https://doi.org/10.1016/S0377-0427(98)00118-6

How to Cite
Zagorodnyuk, S. M. “On the Orthogonality of Partial Sums of the Generalized Hypergeometric Series”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 1, Jan. 2022, pp. 36 -44, doi:10.37863/umzh.v74i1.6989.
Research articles