# contiguous functions

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## 7 matching pages

##### 1: 16.3 Derivatives and Contiguous Functions
###### §16.3(ii) ContiguousFunctions
Two generalized hypergeometric functions ${{}_{p}F_{q}}\left(\mathbf{a};\mathbf{b};z\right)$ are (generalized) contiguous if they have the same pair of values of $p$ and $q$, and corresponding parameters differ by integers. If $p\leq q+1$, then any $q+2$ distinct contiguous functions are linearly related. …
##### 2: 15.5 Derivatives and Contiguous Functions
###### §15.5(ii) ContiguousFunctions
The six functions $F\left(a\pm 1,b;c;z\right)$, $F\left(a,b\pm 1;c;z\right)$, $F\left(a,b;c\pm 1;z\right)$ are said to be contiguous to $F\left(a,b;c;z\right)$. … By repeated applications of (15.5.11)–(15.5.18) any function $F\left(a+k,b+\ell;c+m;z\right)$, in which $k,\ell,m$ are integers, can be expressed as a linear combination of $F\left(a,b;c;z\right)$ and any one of its contiguous functions, with coefficients that are rational functions of $a,b,c$, and $z$. An equivalent equation to the hypergeometric differential equation (15.10.1) is …
##### 3: 16.4 Argument Unity
See Bailey (1964, §§4.3(7) and 7.6(1)) for the transformation formulas and Wilson (1978) for contiguous relations. …
##### 5: 10.21 Zeros
The positive zeros of any two real distinct cylinder functions of the same order are interlaced, as are the positive zeros of any real cylinder function $\mathscr{C}_{\nu}\left(z\right)$ and the contiguous function $\mathscr{C}_{\nu+1}\left(z\right)$. … …
##### 6: Bibliography S
• J. Segura (2008) Interlacing of the zeros of contiguous hypergeometric functions. Numer. Algorithms 49 (1-4), pp. 387–407.
• ##### 7: Bibliography G
• F. Gao and V. J. W. Guo (2013) Contiguous relations and summation and transformation formulae for basic hypergeometric series. J. Difference Equ. Appl. 19 (12), pp. 2029–2042.
• A. Gervois and H. Navelet (1985a) Integrals of three Bessel functions and Legendre functions. I. J. Math. Phys. 26 (4), pp. 633–644.
• A. Gervois and H. Navelet (1985b) Integrals of three Bessel functions and Legendre functions. II. J. Math. Phys. 26 (4), pp. 645–655.
• S. Goldstein (1927) Mathieu functions. Trans. Camb. Philos. Soc. 23, pp. 303–336.
• A. J. Guttmann and T. Prellberg (1993) Staircase polygons, elliptic integrals, Heun functions, and lattice Green functions. Phys. Rev. E 47 (4), pp. R2233–R2236.