33Cxx Hypergeometric functions
This subtopic studies hypergeometric functions, including classical special functions, differential equations, and transformation formulas.
Specific topics
33C05 Classical hypergeometric functions, ${}_2F_1$
Overview
33C05 studies classical hypergeometric functions, ${}_2f_1$ in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for classical hypergeometric functions, ${}_2f_1$
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C10 Bessel and Airy functions, cylinder functions, ${}_0F_1$
Overview
33C10 studies bessel and airy functions, cylinder functions, ${}_0f_1$ in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for bessel and airy functions, cylinder functions, ${}_0f_1$
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C15 Confluent hypergeometric functions, Whittaker functions, ${}_1F_1$
Overview
33C15 studies confluent hypergeometric functions, whittaker functions, ${}_1f_1$ in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for confluent hypergeometric functions, whittaker functions, ${}_1f_1$
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C20 Generalized hypergeometric series, ${}_pF_q$
Overview
33C20 studies generalized hypergeometric series, ${}_pf_q$ in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for generalized hypergeometric series, ${}_pf_q$
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C45 Orthogonal polynomials and functions of hypergeometric type
Overview
33C45 studies orthogonal polynomials and functions of hypergeometric type in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for orthogonal polynomials and functions of hypergeometric type
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C47 Other special orthogonal polynomials and functions
Overview
33C47 studies other special orthogonal polynomials and functions in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for other special orthogonal polynomials and functions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C50 Orthogonal polynomials and functions in several variables
Overview
33C50 studies orthogonal polynomials and functions in several variables in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for orthogonal polynomials and functions in several variables
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C52 Orthogonal polynomials and functions associated with root systems
Overview
33C52 studies orthogonal polynomials and functions associated with root systems in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for orthogonal polynomials and functions associated with root systems
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C55 Spherical harmonics
Overview
33C55 studies spherical harmonics in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for spherical harmonics
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C60 Hypergeometric integrals and functions defined by them
Overview
33C60 studies hypergeometric integrals and functions defined by them in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for hypergeometric integrals and functions defined by them
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C65 Appell, Horn and Lauricella functions
Overview
33C65 studies appell, horn and lauricella functions in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for appell, horn and lauricella functions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C67 Hypergeometric functions associated with root systems
Overview
33C67 studies hypergeometric functions associated with root systems in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for hypergeometric functions associated with root systems
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C70 Other hypergeometric functions and integrals
Overview
33C70 studies other hypergeometric functions and integrals in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for other hypergeometric functions and integrals
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C75 Elliptic integrals as hypergeometric functions
Overview
33C75 studies elliptic integrals as hypergeometric functions in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for elliptic integrals as hypergeometric functions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C80 Connections with groups and algebras
Overview
33C80 studies connections with groups and algebras in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for connections with groups and algebras
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks
33C90 Applications of hypergeometric functions
Overview
33C90 studies applications of hypergeometric functions in hypergeometric functions. It studies hypergeometric and generalized hypergeometric functions, their transformation laws, and differential equations.
Related Wikipedia Page
Hypergeometric Functions (Wikipedia)
Useful Links
Key Ideas
- Canonical formulations and representative examples for applications of hypergeometric functions
- How structural, local, and computational viewpoints interact in the subject
- Standard theorem patterns and invariants used in current research practice
Typical Uses
Used in classical special-function theory and symbolic computation.
Applications
- Hypergeometric series
- Differential equations
- Transformation and summation formulas
References
Recommended Textbooks