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1: 1.12 Continued Fractions
Determinant Formula
2: 1.3 Determinants
Krattenthaler’s Formula
3: Bibliography M
  • T. Masuda, Y. Ohta, and K. Kajiwara (2002) A determinant formula for a class of rational solutions of Painlevé V equation. Nagoya Math. J. 168, pp. 1–25.
  • T. Masuda (2003) On a class of algebraic solutions to the Painlevé VI equation, its determinant formula and coalescence cascade. Funkcial. Ekvac. 46 (1), pp. 121–171.
  • 4: Bibliography B
  • W. Barrett (1981) Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.
  • B. C. Berndt (1975a) Character analogues of the Poisson and Euler-MacLaurin summation formulas with applications. J. Number Theory 7 (4), pp. 413–445.
  • B. C. Berndt (1975b) Periodic Bernoulli numbers, summation formulas and applications. In Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975), pp. 143–189.
  • M. V. Berry and J. P. Keating (1992) A new asymptotic representation for ζ ( 1 2 + i t ) and quantum spectral determinants. Proc. Roy. Soc. London Ser. A 437, pp. 151–173.
  • R. Bo and R. Wong (1999) A uniform asymptotic formula for orthogonal polynomials associated with exp ( x 4 ) . J. Approx. Theory 98, pp. 146–166.
  • 5: 23.10 Addition Theorems and Other Identities
    23.10.5 | 1 ( u ) ( u ) 1 ( v ) ( v ) 1 ( w ) ( w ) | = 0 ,
    §23.10(ii) Duplication Formulas
    §23.10(iii) n -Tuple Formulas
    6: Bibliography V
  • A. J. van der Poorten (1980) Some Wonderful Formulas an Introduction to Polylogarithms. In Proceedings of the Queen’s Number Theory Conference, 1979 (Kingston, Ont., 1979), R. Ribenboim (Ed.), Queen’s Papers in Pure and Appl. Math., Vol. 54, Kingston, Ont., pp. 269–286.
  • R. Vein and P. Dale (1999) Determinants and Their Applications in Mathematical Physics. Applied Mathematical Sciences, Vol. 134, Springer-Verlag, New York.
  • A. Verma and V. K. Jain (1983) Certain summation formulae for q -series. J. Indian Math. Soc. (N.S.) 47 (1-4), pp. 71–85 (1986).
  • 7: 35.7 Gaussian Hypergeometric Function of Matrix Argument
    35.7.6 F 1 2 ( a , b c ; 𝐓 ) = | 𝐈 𝐓 | c a b F 1 2 ( c a , c b c ; 𝐓 ) = | 𝐈 𝐓 | a F 1 2 ( a , c b c ; 𝐓 ( 𝐈 𝐓 ) 1 ) = | 𝐈 𝐓 | b F 1 2 ( c a , b c ; 𝐓 ( 𝐈 𝐓 ) 1 ) .
    Gauss Formula
    Reflection Formula
    8: Bibliography K
  • K. Kajiwara and Y. Ohta (1996) Determinant structure of the rational solutions for the Painlevé II equation. J. Math. Phys. 37 (9), pp. 4693–4704.
  • K. Kajiwara and Y. Ohta (1998) Determinant structure of the rational solutions for the Painlevé IV equation. J. Phys. A 31 (10), pp. 2431–2446.
  • A. A. Kapaev and A. V. Kitaev (1993) Connection formulae for the first Painlevé transcendent in the complex domain. Lett. Math. Phys. 27 (4), pp. 243–252.
  • R. P. Kelisky (1957) On formulas involving both the Bernoulli and Fibonacci numbers. Scripta Math. 23, pp. 27–35.
  • T. H. Koornwinder (1977) The addition formula for Laguerre polynomials. SIAM J. Math. Anal. 8 (3), pp. 535–540.
  • 9: 35.8 Generalized Hypergeometric Functions of Matrix Argument
    Pfaff–Saalschütz Formula
    35.8.13 𝟎 < 𝐗 < 𝐈 | 𝐗 | a 1 1 2 ( m + 1 ) | 𝐈 𝐗 | b 1 a 1 1 2 ( m + 1 ) F q p ( a 2 , , a p + 1 b 2 , , b q + 1 ; 𝐓 𝐗 ) d 𝐗 = 1 B m ( b 1 a 1 , a 1 ) F q + 1 p + 1 ( a 1 , , a p + 1 b 1 , , b q + 1 ; 𝐓 ) , ( b 1 a 1 ) , ( a 1 ) > 1 2 ( m 1 ) .
    10: Errata
  • Paragraph Inversion Formula (in §35.2)

    The wording was changed to make the integration variable more apparent.

  • Section 1.13

    In Equation (1.13.4), the determinant form of the two-argument Wronskian

    1.13.4 𝒲 { w 1 ( z ) , w 2 ( z ) } = det [ w 1 ( z ) w 2 ( z ) w 1 ( z ) w 2 ( z ) ] = w 1 ( z ) w 2 ( z ) w 2 ( z ) w 1 ( z )

    was added as an equality. In ¶Wronskian (in §1.13(i)), immediately below Equation (1.13.4), a sentence was added indicating that in general the n -argument Wronskian is given by 𝒲 { w 1 ( z ) , , w n ( z ) } = det [ w k ( j 1 ) ( z ) ] , where 1 j , k n . Immediately below Equation (1.13.4), a sentence was added giving the definition of the n -argument Wronskian. It is explained just above (1.13.5) that this equation is often referred to as Abel’s identity. Immediately below Equation (1.13.5), a sentence was added explaining how it generalizes for n th-order differential equations. A reference to Ince (1926, §5.2) was added.

  • Usability

    Additional keywords are being added to formulas (an ongoing project); these are visible in the associated ‘info boxes’ linked to the [Uncaptioned image] icons to the right of each formula, and provide better search capabilities.

  • Subsection 14.18(iii)

    This subsection now identifies Equations (14.18.6) and (14.18.7) as Christoffel’s Formulas.

  • Table 18.3.1

    Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.