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Liouville function

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1: 27.2 Functions
27.2.13 λ ( n ) = { 1 , n = 1 , ( - 1 ) a 1 + + a ν ( n ) , n > 1 .
This is Liouville’s function. …
2: Bibliography L
  • L. Lorch, M. E. Muldoon, and P. Szegő (1970) Higher monotonicity properties of certain Sturm-Liouville functions. III. Canad. J. Math. 22, pp. 1238–1265.
  • L. Lorch, M. E. Muldoon, and P. Szegő (1972) Higher monotonicity properties of certain Sturm-Liouville functions. IV. Canad. J. Math. 24, pp. 349–368.
  • L. Lorch and P. Szegő (1963) Higher monotonicity properties of certain Sturm-Liouville functions.. Acta Math. 109, pp. 55–73.
  • 3: 27.6 Divisor Sums
    27.6.1 d | n λ ( d ) = { 1 , n  is a square , 0 , otherwise .
    4: 27.7 Lambert Series as Generating Functions
    27.7.6 n = 1 λ ( n ) x n 1 - x n = n = 1 x n 2 .
    5: Bibliography M
  • M. E. Muldoon (1970) Singular integrals whose kernels involve certain Sturm-Liouville functions. I. J. Math. Mech. 19 (10), pp. 855–873.
  • M. E. Muldoon (1977) Higher monotonicity properties of certain Sturm-Liouville functions. V. Proc. Roy. Soc. Edinburgh Sect. A 77 (1-2), pp. 23–37.
  • 6: 27.4 Euler Products and Dirichlet Series
    27.4.7 n = 1 λ ( n ) n - s = ζ ( 2 s ) ζ ( s ) , s > 1 ,
    7: 1.9 Calculus of a Complex Variable
    Liouville’s Theorem
    8: 15.16 Products
    §15.16 Products
    where A 0 = 1 and A s , s = 1 , 2 , , are defined by the generating function
    Generalized Legendre’s Relation
    For further results of this kind, and also series of products of hypergeometric functions, see Erdélyi et al. (1953a, §2.5.2).
    9: 3.7 Ordinary Differential Equations
    Consideration will be limited to ordinary linear second-order differential equations …where f , g , and h are analytic functions in a domain D . …For applications to special functions f , g , and h are often simple rational functions. …
    §3.7(iv) Sturm–Liouville Eigenvalue Problems
    The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the system …
    10: Bibliography D
  • T. M. Dunster, D. A. Lutz, and R. Schäfke (1993) Convergent Liouville-Green expansions for second-order linear differential equations, with an application to Bessel functions. Proc. Roy. Soc. London Ser. A 440, pp. 37–54.