# symbolic operations

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## 1—10 of 33 matching pages

##### 2: 34.9 Graphical Method
The graphical method establishes a one-to-one correspondence between an analytic expression and a diagram by assigning a graphical symbol to each function and operation of the analytic expression. …
##### 3: Guide to Searching the DLMF
Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators:
• term:

a textual word, a number, or a math symbol.

• Boolean operator:

and and or

• proximity operator:

adj, prec/n, and near/n, where n is any positive natural number.

##### 4: 17.6 ${{}_{2}\phi_{1}}$ Function
17.6.25 $\mathcal{D}_{q}^{n}{{}_{2}\phi_{1}}\left({a,b\atop c};q,zd\right)=\frac{\left(% a,b;q\right)_{n}d^{n}}{\left(c;q\right)_{n}(1-q)^{n}}{{}_{2}\phi_{1}}\left({aq% ^{n},bq^{n}\atop cq^{n}};q,dz\right),$
17.6.26 $\mathcal{D}_{q}^{n}\left(\frac{\left(z;q\right)_{\infty}}{\left(abz/c;q\right)% _{\infty}}{{}_{2}\phi_{1}}\left({a,b\atop c};q,z\right)\right)=\frac{\left(c/a% ,c/b;q\right)_{n}}{\left(c;q\right)_{n}(1-q)^{n}}\left(\frac{ab}{c}\right)^{n}% \frac{\left(zq^{n};q\right)_{\infty}}{\left(abz/c;q\right)_{\infty}}{{}_{2}% \phi_{1}}\left({a,b\atop cq^{n}};q,zq^{n}\right).$
##### 5: 18.42 Software
Also included is a website (CAOP) operated by a university department. … A more complete list of available software for computing these functions, and for generating formulas symbolically, is found in the Software Index. …
##### 6: 17.2 Calculus
17.2.42 $f^{[n]}(z)=\mathcal{D}_{q}^{n}f(z)=\begin{cases}z^{-n}(1-q)^{-n}\sum_{j=0}^{n}% q^{-nj+\genfrac{(}{)}{0.0pt}{}{j+1}{2}}(-1)^{j}\genfrac{[}{]}{0.0pt}{}{n}{j}_{% q}f(zq^{j}),&z\neq 0,\\ \dfrac{f^{(n)}(0)\left(q;q\right)_{n}}{n!(1-q)^{n}},&z=0.\end{cases}$
##### 7: 26.22 Software
Also included are websites operated by university departments and consortia, research institutions, and peer-reviewed journals. …
• Inverse Symbolic Calculator (website).

As a general rule, using the latest available version of your chosen browser, plugins and an updated operating system is helpful. … Since the display of mathematics involves many special symbols not often seen in plain text, a MathML renderer generally needs special fonts. … For other browsers, you may see a ? or a box like indicating missing symbols, and thus insufficient fonts. …
##### 9: Bibliography R
• J. Raynal (1979) On the definition and properties of generalized $6$-$j$ symbols. J. Math. Phys. 20 (12), pp. 2398–2415.
• REDUCE (free interactive system)
• S. Ritter (1998) On the computation of Lamé functions, of eigenvalues and eigenfunctions of some potential operators. Z. Angew. Math. Mech. 78 (1), pp. 66–72.
• C. C. J. Roothaan and S. Lai (1997) Calculation of $3n$-$j$ symbols by Labarthe’s method. International Journal of Quantum Chemistry 63 (1), pp. 57–64.
• J. Rushchitsky and S. Rushchitska (2000) On Simple Waves with Profiles in the form of some Special Functions—Chebyshev-Hermite, Mathieu, Whittaker—in Two-phase Media. In Differential Operators and Related Topics, Vol. I (Odessa, 1997), Operator Theory: Advances and Applications, Vol. 117, pp. 313–322.
• ##### 10: Errata
• Notation

In §3.7(iii), the symbol $\mathbf{A}_{P}$ is now being used in several places instead of $\mathbf{A}$ in order to disambiguate symbols.

• Equation (3.3.34)

In the online version, the leading divided difference operators were previously omitted from these formulas, due to programming error.

Reported by Nico Temme on 2021-06-01

Pochhammer and $q$-Pochhammer symbols in several equations now correctly link to their definitions.
A number of changes were made with regard to fractional integrals and derivatives. In §1.15(vi) a reference to Miller and Ross (1993) was added, the fractional integral operator of order $\alpha$ was more precisely identified as the Riemann-Liouville fractional integral operator of order $\alpha$, and a paragraph was added below (1.15.50) to generalize (1.15.47). In §1.15(vii) the sentence defining the fractional derivative was clarified. In §2.6(iii) the identification of the Riemann-Liouville fractional integral operator was made consistent with §1.15(vi).