Hi, In the `numpy.polynomial.chebyshev` module, the function for raising a Chebyshev polynomial to a power, `chebpow` [1], is essentially implemented in the following way: {{{#!highlight python def chebpow(c, pow): """Raise a Chebyshev series to a power.""" zs = _cseries_to_zseries(c) prd = zs for i in range(2, pow + 1): prd = np.convolve(prd, zs) return _zseries_to_cseries(prd) }}} For large coefficient arrays `c` and big exponents `pow`, this procedure is not efficient. In fact, the complexity of this function is `O(pow*len(c)^2)`, since the numpy convolution does not make use of a Fast Fourier Transform (FFT). It is known that Chebyshev polynomials can be multiplied with Discrete Cosine Transforms (DCT) [2]. What results is the following algorithm`O(pow*len(c)*log(pow*len(c)))` algorithm for raising a Chebyshev polynomial with coefficients `c` to an integer power: {{{#!highlight python def chebpow_dct(c, pow): """Raise a Chebyshev series to a power.""" pad_length = (pow - 1) * (len(c) - 1) c = np.pad(c, (0, pad_length)) c[1:-1] /= 2 c_pow = idct(dct(c) ** pow) c_pow[1:-1] *= 2 return c_pow }}} The only issue I am having is that as far as I know, `numpy` (unlike `scipy`) does not have a specialized implementation for the DCT. So the only way of getting the code to work is "emulating" a DCT with two calls to `numpy.fft.rfft`, which is slightly slower than using the `scipy.fft.dct`. I have created a Google colab notebook which compares the error and runtime of the different implementations (current implementation, implementation using `scipy.fft.dct`, and pure numpy implementation) [3]. Especially for larger degree polynomials and higher powers this enhancement would make a huge difference in terms of runtime. Similarly, `chebmul` and `chebinterpolate` can also be implemented more efficiently by using a DCT. Do you think this enhancement is worth pursuing, and should I create a pull-request for it? Best, Fabio [1] https://github.com/numpy/numpy/blob/main/numpy/polynomial/chebyshev.py#L817 [2] https://www.sciencedirect.com/science/article/pii/0024379595006966 [3] https://colab.research.google.com/drive/1JtDDeWC1CEQHDidZ9f5_Ma_ifoBv4Tuz?us...